Let $\alpha \in C_k^d$. Let $\tilde{\alpha} \stackrel{\text{def.}}{=} (\alpha_2, \dots, \alpha_d) \in \mathbb{N}^{d-1}$. Then, from Lemma 1488 (indices are smaller than maximal length), and Definition 1486 (sets of multi-indices $A_k^d$ and $C_k^d$), we have $0 \le \alpha_1 \le k$, $\tilde{\alpha} \in C_{k-\alpha_1}^{d-1}$, and $\alpha = (\alpha_1, \tilde{\alpha}) \in \tilde{S}_{k,\alpha_1}^d$. Thus, we have $C_k^d \supset \bigcup_{i=0}^k \tilde{S}_{k,i}^d$.

Therefore, we have $C_k^d = \bigcup_{i=0}^k \tilde{S}_{k,i}^d$.

**Case of $\tilde{S}_{k,i}^d$.** The proof is very similar. Let $\alpha \in \tilde{S}_{k,i}^d$. Then, from Definition 1491 (slices $\tilde{S}_{k,i}^d$ and $\tilde{S}_{k,j}^d$), and Definition 1486 (sets of multi-indices $A_k^d$ and $C_k^d$), there exists $\tilde{\alpha} \in C_{k-i}^{d-1}$ such that $\alpha = (\tilde{\alpha}, i)$, and we have $|\alpha| = |\tilde{\alpha}| + i = k - i + i = k$, i.e. $\alpha \in C_k^d$, and $\tilde{S}_{k,i}^d \subset C_k^d$. Thus, we have $\bigcup_{i=0}^k \tilde{S}_{k,i}^d \subset C_k^d$.

Let $i, j \in [0..k]$. Let $\alpha \in \tilde{S}_{k,i}^d \cap \tilde{S}_{k,j}^d$. Then, from Definition 1491 (slices $\tilde{S}_{k,i}^d$ and $\tilde{S}_{k,j}^d$), there exist $\tilde{\alpha}^i \in C_{k-i}^{d-1}$ and $\tilde{\alpha}^j \in C_{k-j}^{d-1}$ such that $\alpha = (\tilde{\alpha}_i, i) = (\tilde{\alpha}_j, j)$. Thus, we have $\tilde{\alpha}_i = \tilde{\alpha}_j$ and $i = j$, and by contrapositive, we have $\tilde{S}_{k,i}^d \cap \tilde{S}_{k,j}^d = \emptyset$ when $i \ne j$.

Let $\alpha \in C_k^d$. Let $\tilde{\alpha} \stackrel{\text{def.}}{=} (\alpha_1, \dots, \alpha_{d-1}) \in \mathbb{N}^{d-1}$. Then, from Lemma 1488 (indices are smaller than maximal length), and Definition 1486 (sets of multi-indices $A_k^d$ and $C_k^d$), we have $0 \le \alpha_d \le k$, $\tilde{\alpha} \in C_{k-\alpha_d}^{d-1}$, and $\alpha = (\tilde{\alpha}, \alpha_d) \in \tilde{S}_{k,\alpha_d}^d$. Thus, we have $C_k^d \supset \bigcup_{i=0}^k \tilde{S}_{k,i}^d$.

Therefore, we have $C_k^d = \bigcup_{i=0}^k \tilde{S}_{k,i}^d$.

**Lemma 1494 (cardinal of slices of $C_k^d$).** Let $d \ge 2$. Let $k \in \mathbb{N}$. Let $i \in [0..k]$.

Let $\tilde{\varphi}_{k,d}^i \stackrel{\text{def.}}{=} (\alpha = (i, \tilde{\alpha}) \in \tilde{S}_{k,i}^d \mapsto \tilde{\alpha} \in C_{k-i}^{d-1})$, and $\tilde{\varphi}_{k,d}^i \stackrel{\text{def.}}{=} (\alpha = (\tilde{\alpha}, i) \in \tilde{S}_{k,i}^d \mapsto \tilde{\alpha} \in C_{k-i}^{d-1})$.

Then, $\tilde{\varphi}_{k,d}^i$ and $\tilde{\varphi}_{k,d}^i$ are bijections, and we have $\text{card}(\tilde{S}_{k,i}^d) = \text{card}(\tilde{S}_{k,i}^d) = \text{card}(C_{k-i}^{d-1})$.

**Proof.** From Definition 1491 (slices $\tilde{S}_{k,i}^d$ and $\tilde{S}_{k,j}^d$), the applications $\tilde{\varphi}_{k,d}^i$ and $\tilde{\varphi}_{k,d}^i$ are well defined, and obviously surjective. Moreover, let $\alpha, \beta \in \tilde{S}_{k,i}^d$, such that $\tilde{\alpha} = \tilde{\varphi}_{k,d}^i(\alpha) = \tilde{\varphi}_{k,d}^i(\beta) = \tilde{\beta}$. Thus, we have $(i, \tilde{\alpha}) = (i, \tilde{\beta})$, and $\tilde{\varphi}_{k,d}^i$ is injective. In the same manner, let $\alpha, \beta \in \tilde{S}_{k,i}^d$, such that $\tilde{\alpha} = \tilde{\varphi}_{k,d}^i(\alpha)$ equals $\tilde{\varphi}_{k,d}^i(\beta) = \tilde{\beta}$. Thus, we have $(\tilde{\alpha}, i) = (\tilde{\beta}, i)$, and $\tilde{\varphi}_{k,d}^i$ is injective.

Therefore, $\tilde{\varphi}_{k,d}^i$ and $\tilde{\varphi}_{k,d}^i$ are bijective, and we have the result.

**Lemma 1495 (cardinal of $C_k^d$).** Let $d \ge 1$. Let $k \in \mathbb{N}$. Then, the number of elements of $C_k^d$ is

$$ (9.10) \qquad \text{card}(C_k^d) = \binom{k+d-1}{d-1} = \binom{k+d-1}{k} = \frac{(k+d-1)!}{k!(d-1)!}. $$

**Proof.** From Lemma 1364 (properties of the binomial coefficient, (5.6)), and Definition 1362 (binomial coefficient), we have the last two equalities.

**First equality.** For all $d \in \mathbb{N}^*$, let $P(d) \stackrel{\text{def.}}{=} [\text{card}(C_k^d) = \binom{k+d-1}{d-1}]$.

**Induction: $P(1)$.** Direct consequence of Lemma 1489 (first $C_k^d$, (9.4)), and Lemma 1364 (properties of the binomial coefficient, (5.4)).

**Induction: $P(d) \Rightarrow P(d+1)$.**

Assume that $P(d)$ holds. Then, from Lemma 1493 (slices of $C_k^d$), Lemma 1494 (cardinal of slices of $C_k^d$), and Lemma 1364 (properties of the binomial coefficient, (5.8)), we have

$$ \begin{aligned} \text{card}(C_k^{d+1}) &= \sum_{i=0}^{k} \text{card}(\tilde{S}_{k,i}^{d+1}) = \sum_{i=0}^{k} \text{card}(C_{k-i}^{d}) \\ &= \sum_{i=0}^{k} \binom{k-i+d-1}{d-1} = \sum_{j=0}^{k} \binom{j+d-1}{d-1} = \binom{k+d}{d}. \end{aligned} $$

This concludes the induction on $d$, and we have for all $d \ge 1$, $P(d)$.
### 9.1.4 Layers of $\mathcal{A}_k^d$ and its cardinal

**Lemma 1496 (C_k^d are layers of A_k^d).**
Let $d \ge 1$. Let $k \in \mathbb{N}$. Then, we have

$$ (9.11) \qquad \mathcal{A}_0^d = \mathcal{C}_0^d \quad \text{and} \quad \mathcal{A}_{k+1}^d = \mathcal{A}_k^d \uplus \mathcal{C}_{k+1}^d, $$

thus, $\mathcal{A}_k^d = \biguplus_{l=0}^k \mathcal{C}_l^d$ and the sequence $(\mathcal{A}_k^d)_{k \in \mathbb{N}}$ is increasing.

*Proof.* Direct consequence of Definition 1486 (sets of multi-indices $\mathcal{A}_k^d$ and $\mathcal{C}_k^d$), and **the definition of (disjoint) union.** $\square$

**Lemma 1497 (first multi-indices $\mathcal{A}_k^d$).**
Let $d \ge 1$. Let $k \in \mathbb{N}$. Then, we have

$$ (9.12) \qquad (d=1) \qquad \mathcal{A}_k^1 = [0..k], \quad \text{and} \quad \text{card}(\mathcal{A}_k^1) = k+1, $$

$$ (9.13) \qquad (k=0) \qquad \mathcal{A}_0^d = \{0 \in \mathbb{N}^d\}, \quad \text{and} \quad \text{card}(\mathcal{A}_0^d) = 1, $$

$$ (9.14) \qquad (k=1) \qquad \mathcal{A}_1^d = \{0, e_1, \dots, e_d\}, \quad \text{and} \quad \text{card}(\mathcal{A}_1^d) = d+1. $$

*Proof.*

Direct consequence of Definition 1496 ($\mathcal{C}_k^d$ are layers of $\mathcal{A}_k^d$), and Definition 1489 (first $\mathcal{C}_k^d$). $\square$

**Lemma 1498 (cardinal of $\mathcal{A}_k^d$).** Let $d \ge 1$. Let $k \in \mathbb{N}$. Then, the number of elements of $\mathcal{A}_k^d$ is

$$ (9.15) \qquad \text{card}(\mathcal{A}_k^d) = \binom{k+d}{d} = \binom{k+d}{k} = \frac{(k+d)!}{k! d!}. $$

By extension, for $d=0$, we also set $\text{card}(\mathcal{A}_k^0) \stackrel{\text{def.}}{=} \binom{k}{0} = \binom{k}{0} = 1$.

*Proof.* Direct consequence of Lemma 1496 ($\mathcal{C}_k^d$ are layers of $\mathcal{A}_k^d$, thus, $\text{card}(\mathcal{A}_k^d) = \sum_{l=0}^k \text{card}(\mathcal{C}_l^d)$), Lemma 1495 (cardinal of $\mathcal{C}_k^d$), Lemma 1364 (properties of the binomial coefficient, (5.8)), and Lemma 1364 (properties of the binomial coefficient, (5.6), (5.4)), thus last two equalities hold, and with $d=0$ too). $\square$

### 9.1.5 Other cardinals

*Remark 1499.* In Figure 9.3, we plot examples of the mappings $f_{k,0}^d$ and $f_{k,1}^d$, defined over multi-index sets in Lemmas 1500 and 1501.

These functions are used for the transfer of nodes from a $(d-1)$-simplex to the hyperface of a $d$-simplex, see Lemmas 1605 and 1607. The transfer is made via the geometric applications $\phi_{0^{d-1}i}^{[\mathbf{v}]^d}$ (for $i \in [0..d]$), that map an hyperface to a simplex, see Lemma 1584, and Figure 9.6.

**Lemma 1500 (cardinal of $\mathcal{C}_k^d$ and $\mathcal{A}_k^{d-1}$).**
Let $d \ge 1$. Let $k \in \mathbb{N}$.

Let $f_{k,0}^d \stackrel{\text{def.}}{=} (\tilde{\alpha} \in \mathcal{A}_k^{d-1} \mapsto (k - |\tilde{\alpha}|, \tilde{\alpha}) \in \mathcal{C}_k^d)$, and $\tilde{f}_{k,0}^d \stackrel{\text{def.}}{=} (\tilde{\alpha} \in \mathcal{A}_k^{d-1} \mapsto (\tilde{\alpha}, k - |\tilde{\alpha}|) \in \mathcal{C}_k^d)$.

Then, $f_{k,0}^d$ and $\tilde{f}_{k,0}^d$ are bijections, and we have $\text{card}(\mathcal{C}_k^d) = \text{card}(\mathcal{A}_k^{d-1})$.

*Proof. Identity on the cardinals.*

Direct consequence of Lemma 1495 (cardinal of $\mathcal{C}_k^d$), and Lemma 1498 (cardinal of $\mathcal{A}_k^d$).

**Well-defined applications.** Let $\tilde{\alpha}, \tilde{\alpha} \in \mathcal{A}_k^{d-1}$. Then, from Definition 1486 (sets of multi-indices $\mathcal{A}_k^d$ and $\mathcal{C}_k^d$), and Definition 1480 (length of multi-indices), we have $k - |\tilde{\alpha}|, k - |\tilde{\alpha}| \ge 0$, and $|f_{k,0}^d(\tilde{\alpha})| = (k - |\tilde{\alpha}|) + |\tilde{\alpha}| = k = |\tilde{\alpha}| + (k - |\tilde{\alpha}|) = |\tilde{f}_{k,0}^d(\tilde{\alpha})|$. Thus, $f_{k,0}^d$ and $\tilde{f}_{k,0}^d$ are well defined.

**Bijections.** Direct consequence of the definition of $f_{k,0}^d$ and $\tilde{f}_{k,0}^d$ (obviously injective), and **the fact that injectivity and cardinal equality imply bijectivity.** $\square$
<figure><img src="image_21.png" /><figcaption>Figure 9.3: Mappings $f_{k,0}^d$ and $f_{k,1}^d$ in the case $d=k=3$ (see Lemmas 1500 and 1501).</figcaption></figure>

The multi-indices in $\mathcal{A}_3^2$ are mapped to multi-indices of $C_3^3$ (for $f_{k,0}^d$) or to multi-indices of $\mathcal{A}_{3,1}^3$ (for $f_{k,1}^d$). This is illustrated geometrically with the representation of the nodes in the triangle or the tetrahedra.

For $f_{k,0}^d$, the reference triangle nodes are mapped onto the nodes of the blue face of the tetrahedron. This face, opposite vertex $\hat{v}_0$, contains the nodes having indices in $C_k^d$. The coloring of the nodes is intended to help see the mapping: for all $(i,j) \in \mathcal{A}_2^3$, we have $f_{k,0}^d(i,j) = (3-(i+j), i,j)$.

For $f_{k,1}^d$, the reference triangle nodes are mapped onto the nodes of the magenta face of the tetrahedron. This face, opposite vertex $\hat{v}_1$, contains the nodes having indices in $\mathcal{A}_{k,1}^d$. For all $(i,j) \in \mathcal{A}_2^3$, we have $f_{k,1}^d(i,j) = (0, i,j)$.

**Lemma 1501 (cardinal of $\mathcal{A}_{k,i}^d$ and $\mathcal{A}_k^{d-1}$).**

Let $d \ge 1$. Let $k \in \mathbb{N}$. Let $i \in [1..d]$.

Let $f_{k,i}^d \stackrel{\text{def.}}{=} (\alpha' \in \mathcal{A}_k^{d-1} \mapsto (\alpha'_1, \dots, \alpha'_{i-1}, 0, \alpha'_i, \dots, \alpha'_{d-1}) \in \mathcal{A}_{k,i}^d)$.

Then, $f_{k,i}^d$ is a bijection that preserves length, and we have $\text{card}(\mathcal{A}_{k,i}^d) = \text{card}(\mathcal{A}_k^{d-1})$.

**Proof. (1) Well-defined application and preservation of length.**

Let $\alpha' \in \mathcal{A}_k^{d-1}$. Then, from Definition 1480 (length of multi-indices), and Definition 1486 (sets of multi-indices $\mathcal{A}_k^d$ and $C_k^d$), we have $|f_{k,i}^d(\alpha')| = |\alpha'| \le k$, and $f_{k,i}^d(\alpha') \in \mathcal{A}_{k,i}^d$.

(2) **Injectivity.** Direct consequence of the definition of $f_{k,i}^d$.

(3) **Surjectivity.**

Let $\alpha \in \mathcal{A}_{k,i}^d$. Let $\alpha' \stackrel{\text{def.}}{=} (\alpha_1, \dots, \alpha_{i-1}, \alpha_{i+1}, \dots, \alpha_d) \in \mathbb{N}^{d-1}$. Then, from (1), and Definition 1486 (sets of multi-indices $\mathcal{A}_k^d$ and $C_k^d$), we have $f_{k,i}^d(\alpha') = \alpha$ and $|\alpha'| = |\alpha| \le k$. Thus, $\alpha' \in \mathcal{A}_k^{d-1}$.
Therefore, from (2), (3), **the definition of bijectivity**, and **the definition of cardinal**, $f_{k,i}^d$ is bijective, and we have the equality of cardinals. □

## 9.2 Multivariate polynomials

*Remark 1502.* See also univariate polynomials in Section 5.6.

In the following statements, $\mathbb{P}_k^d$ is defined as the space of polynomials of total degree at most $k$. Note that the notion of degree is fully defined *afterwards*.

### 9.2.1 Monomials and polynomials of $\mathbb{P}_k^d$

**Definition 1503 (monomial in $d$ variables).** Let $d \ge 1$. Let $k \in \mathbb{N}$. Let $\alpha \in \mathbb{C}_k^d$. The monomial of degree $k$ in $d$ variables of multi-exponent $\alpha$ is denoted $\mathbf{X}^\alpha$, and is defined by

$$ (9.16) \qquad \mathbf{X}^{\boldsymbol{\alpha}} \stackrel{\text{def.}}{=} \left( \mathbf{x} \in \mathbb{R}^d \mapsto \prod_{i=1}^d x_i^{\alpha_i} \in \mathbb{R} \right). $$

Moreover, 1 is a shortcut for $\mathbf{X}^0$ (the constant function of value 1, which may be omitted in a multiplicative context), and for all $i \in [1..d]$, $X_i$ is a shortcut for $\mathbf{X}^{e_i} = (\mathbf{x} \mapsto x_i)$.

**Lemma 1504 (monomial in $d$ variables for $d=1$ is monomial of a single variable).**

Let $k \in \mathbb{N}$. Then, $\mathbf{X}^\alpha$ for $\alpha \in \mathbb{C}_k^d$ and $d \stackrel{\text{def.}}{=} 1$, and $\mathbf{X}^k$ from Definition 1416 coincide.

*Proof.*

Direct consequence of Definition 1503 (monomial in $d$ variables, (9.16) with $d \stackrel{\text{def.}}{=} 1$), Lemma 1489 (first $\mathbb{C}_k^d$, with $d \stackrel{\text{def.}}{=} 1$), and Definition 1416 (monomial of a single variable, with $\alpha \stackrel{\text{def.}}{=} k$). □

**Definition 1505 (polynomial space $\mathbb{P}_k^d$).** Let $d \ge 1$. Let $k \in \mathbb{N}$. The space of polynomials of degree at most $k$ of $d$ variables is denoted $\mathbb{P}_k^d$, and is defined by

$$ (9.17) \qquad \mathbb{P}_k^d \stackrel{\text{def.}}{=} \operatorname{span} (\mathbf{X}^\alpha)_{\alpha \in \mathcal{A}_k^d} = \left\{ \left( \mathbf{x} \mapsto \sum_{\alpha \in \mathcal{A}_k^d} a_\alpha x_1^{\alpha_1} \dots x_d^{\alpha_d} \right) : \mathbb{R}^d \to \mathbb{R} \middle| (a_\alpha)_{\alpha \in \mathcal{A}_k^d} \in \mathbb{R} \right\}. $$

**Lemma 1506 (polynomial space $\mathbb{P}_k^d$ for $d=1$ is $\mathbb{P}_k^1$).**

Let $k \in \mathbb{N}$. Then, $\mathbb{P}_k^d$ for $d \stackrel{\text{def.}}{=} 1$, and $\mathbb{P}_k^1$ from Definition 1417 coincide.

*Proof.* Direct consequence of Definition 1505 (polynomial space $\mathbb{P}_k^d$, (9.17) with $d \stackrel{\text{def.}}{=} 1$), Lemma 1497 (first multi-indices $\mathcal{A}_k^d$, with $d \stackrel{\text{def.}}{=} 1$), and Definition 1417 (polynomial space $\mathbb{P}_k^1$). □

**Lemma 1507 ($\mathbb{P}_k^d$ is vector space).** Let $d \ge 1$. Let $k \in \mathbb{N}$. Then, $\mathbb{P}_k^d$ is a vector space.

*Proof.* Direct consequence of Definition 1505 (polynomial space $\mathbb{P}_k^d$), and **the definition of the linear span**. □

**Lemma 1508 ($\mathbb{P}_k^d$ is nondecreasing sequence in $k$).**

Let $d \ge 1$. Then, the sequence $(\mathbb{P}_k^d)_{k \in \mathbb{N}}$ is nondecreasing for the inclusion.

*Proof.* Direct consequence of Definition 1505 (polynomial space $\mathbb{P}_k^d$), Lemma 1496 ($\mathbb{C}_k^d$ are layers of $\mathcal{A}_k^d$, increasing sequence), and **monotonicity of the linear span**. □
**Lemma 1509 (constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$).**

Let $d \ge 1$. The spaces of polynomials of $d$ variables of degree at most 0 and 1 are respectively the spaces of constant and affine maps,

$$ (9.18) \qquad \mathbb{P}_0^d = \operatorname{span}(1) = \{(\mathbf{x} \mapsto a_0) \mid a_0 \in \mathbb{R}\}, $$

$$ (9.19) \qquad \begin{aligned} \mathbb{P}_1^d &= \operatorname{span}(1, X_1, X_2, \dots, X_d) \\ &= \{(\mathbf{x} \mapsto a_0 + a_{e_1}x_1 + \dots + a_{e_d}x_d) \mid a_0, a_{e_1}, \dots, a_{e_d} \in \mathbb{R}\} \\ &= \operatorname{Aff}(\mathbb{R}^d, \mathbb{R}). \end{aligned} $$

Thus, $p \in \mathbb{P}_1^d$ iff there exists $a_0 \in \mathbb{R}$ and $A \in \mathcal{M}_{1,d}(\mathbb{R})$ such that, for all $\mathbf{x} \in \mathbb{R}^d$, $p(\mathbf{x}) = a_0 + Ax$.

*Proof.* Direct consequence of Definition 1505 (*polynomial space* $\mathbb{P}_k^d$), Definition 1365 (*canonic families*), Lemma 1497 (*first multi-indices* $\mathcal{A}_k^d$), Lemma 1399 (*equivalent definition of affine map (finite dimension)*). $\square$

**Definition 1510 (degree of polynomial).** Let $d \ge 1$. Let $k \in \mathbb{N}$. Let $(a_\alpha)_{\alpha \in \mathcal{A}_k^d} \in \mathbb{R}$.

Let $p \stackrel{\text{def.}}{=} \sum_{\alpha \in \mathcal{A}_k^d} a_\alpha \mathbf{X}^\alpha \in \mathbb{P}_k^d$. The degree of $p$ is denoted $\deg p$, and is defined by

$$ (9.20) \qquad \deg p \stackrel{\text{def.}}{=} \max(|\alpha|)_{a_\alpha \neq 0}, $$

with the convention that the maximum of an empty family is $-\infty$, i.e. when $p \stackrel{\text{def.}}{=} 0$.

**Lemma 1511 (values of degree of polynomial).**

Let $d \ge 1$. Let $k \in \mathbb{N}$. Let $p \in \mathbb{P}_k^d$. Then, we have $\deg p \in \{-\infty\} \cup [0..k]$.

*Proof.* Direct consequence of Definition 1510 (*degree of polynomial*). $\square$

**Lemma 1512 (monomials of $C_k^d$ have degree $k$).**

Let $d \ge 1$. Let $k \in \mathbb{N}$. Let $\alpha \in C_k^d$. Then, we have $\deg \mathbf{X}^\alpha = |\alpha| = k$.

*Proof.* Direct consequence of Definition 1510 (*degree of polynomial*), Definition 1503 (*monomial in $d$ variables*), Definition 1486 (*sets of multi-indices* $\mathcal{A}_k^d$ and $C_k^d$). $\square$

**Lemma 1513 ($\mathbb{P}_k^d$ is space of degree at most $k$).** Let $d \ge 1$.

Let $\mathcal{P}(\mathbb{R}^d)$ be the infinite-dimensional space of polynomials of $d$ variables on $\mathbb{R}$. Let $k \in \mathbb{N}$. Then, $\mathbb{P}_k^d$ is the space of polynomials of degree at most $k$,

$$ (9.21) \qquad \mathbb{P}_k^d = \{p \in \mathcal{P}(\mathbb{R}^d) \mid \deg p \le k\}. $$

*Proof.* Direct consequence of Definition 1505 (*polynomial space* $\mathbb{P}_k^d$), Definition 1510 (*degree of polynomial*), and Lemma 1496 ($C_k^d$ are layers of $\mathcal{A}_k^d$). $\square$

### 9.2.2 Product of polynomials

**Lemma 1514 (product of monomials).**

Let $d \ge 1$. Let $k, l \in \mathbb{N}$. Let $\alpha \in C_k^d$ and $\beta \in C_l^d$. Then, we have

$$ (9.22) \qquad \mathbf{X}^{\alpha} \mathbf{X}^{\beta} = \mathbf{X}^{\alpha+\beta}, \quad \text{and} \quad \deg(\mathbf{X}^{\alpha} \mathbf{X}^{\beta}) = \deg \mathbf{X}^{\alpha} + \deg \mathbf{X}^{\beta} = |\alpha| + |\beta| = k+l. $$

*Proof.* Direct consequence of Definition 1503 (*monomial in $d$ variables*), commutative ring properties of $\mathbb{R}$ (thus $\prod_{i=1}^d x_i^{\alpha_i} \times \prod_{i=1}^d x_i^{\beta_i} = \prod_{i=1}^d x_i^{\alpha_i} x_i^{\beta_i}$), Lemma 1512 (*monomials of $C_k^d$ have degree $k$*), and Lemma 1481 (*length of multi-indices is additive*). $\square$
**Lemma 1515 (product of monomial and polynomial).**

Let $d \ge 1$. Let $k, l \in \mathbb{N}$. Let $\alpha \in \mathcal{A}_k^d$. Let $q \in \mathbb{P}_l^d$. Then, we have $\mathbf{X}^{\alpha} q \in \mathbb{P}_{k+l}^d$.

*Proof.* From Definition 1505 (*polynomial space* $\mathbb{P}_k^d$), *commutative ring properties of R*, Lemma 1514 (*product of monomials*), and Definition 1486 (*sets of multi-indices* $\mathcal{A}_k^d$ and $\mathcal{C}_k^d$, thus $|\alpha| \le k$ and $|\beta| \le l$), there exists $(b_\beta)_{\beta \in \mathcal{A}_l^d} \in \mathbb{R}$ such that

$$ \mathbf{X}^{\alpha} q = \mathbf{X}^{\alpha} \left( \sum_{\beta \in \mathcal{A}_l^d} b_{\beta} \mathbf{X}^{\beta} \right) = \sum_{\beta \in \mathcal{A}_l^d} b_{\beta} \mathbf{X}^{\alpha+\beta}, \quad \text{and} \quad |\alpha + \beta| \le k+l. $$

Thus, from Definition 1505 (*polynomial space* $\mathbb{P}_k^d$), we have $\mathbf{X}^{\alpha} q \in \mathbb{P}_{k+l}^d$. $\square$

**Lemma 1516 (product of two polynomials).**

Let $d \ge 1$. Let $k, l \in \mathbb{N}$. Let $p \in \mathbb{P}_k^d$ and $q \in \mathbb{P}_l^d$. Then, we have $pq \in \mathbb{P}_{k+l}^d$.

*Proof.* From Definition 1505 (*polynomial space* $\mathbb{P}_k^d$), and **ring properties of R**, there exists coefficients $(a_\alpha)_{\alpha \in \mathcal{A}_k^d} \in \mathbb{R}$ such that $pq = \sum_{\alpha \in \mathcal{A}_k^d} a_\alpha (\mathbf{X}^\alpha q)$. Thus, from Lemma 1515 (*product of monomial and polynomial*), and Lemma 1507 (*$\mathbb{P}_k^d$ is vector space*), we have $pq \in \mathbb{P}_{k+l}^d$. $\square$

### 9.2.3 Linear independence of monomials

**Lemma 1517 (partial derivative of monomials).**

Let $d \ge 1$. Let $k, l \in \mathbb{N}$.

Let $\alpha \in \mathcal{A}_k^d$ and $\beta \in \mathcal{A}_l^d$. Then, the partial derivative of order $\beta$ of the monomial $\mathbf{X}^{\alpha}$ is

$$ (9.23) \qquad \partial^{\beta} \mathbf{X}^{\alpha} = \begin{cases} \prod_{i=1}^{d} \left( \prod_{j=0}^{\beta_i - 1} (\alpha_i - j) \right) X_i^{\alpha_i - \beta_i} & \text{when } \forall i \in [1..d], \beta_i \le \alpha_i, \\ 0 & \text{otherwise.} \end{cases} $$

Thus, $\partial^{\beta} \mathbf{X}^{\alpha} \in \mathbb{P}_k^d$.

Moreover, if for all $i \in [1..d]$, $\beta_i \le \alpha_i$, then we have $\deg(\partial^{\beta} \mathbf{X}^{\alpha}) = |\alpha| - |\beta| (= k - l)$.

*Proof.* Direct consequence of Definition 1503 (*monomial in d variables*), Lemma 1512 (*monomials of $\mathcal{C}_k^d$ have degree k*), and **the differentiation rules** for univariate polynomials,

$$ \forall i \in [1..d], \quad \frac{\partial^{\beta_i} (x_i^{\alpha_i})}{\partial x_i^{\beta_i}} = \begin{cases} x_i^{\alpha_i} & \text{when } \beta_i = 0, \\ \left( \prod_{j=0}^{\beta_i - 1} (\alpha_i - j) \right) x_i^{\alpha_i - \beta_i} & \text{when } 0 < \beta_i \le \alpha_i, \\ 0 & \text{when } \alpha_i < \beta_i. \end{cases} $$

$\square$

**Remark 1518**

Note that in the first clause of (9.23), we use the convention that a product indexed by the empty set equals 1, the identity element for multiplication, i.e. when some $\beta_i$ is zero.

Note also that the second clause may be omitted, provided the use of the convention that subtraction is closed in $\mathbb{N}$ (i.e. $n - p = 0$ when $n < p$). Indeed, when there exists some $i \in [1..d]$ such that $\alpha_i < \beta_i$, then $x_i^{\alpha_i - \beta_i} = 1$ and $\prod_{j=0}^{\beta_i - 1} (\alpha_i - j) = 0$.
**Lemma 1519 (partial derivative is linear).**
Let $d \ge 1$. Let $k, l \in \mathbb{N}$. Let $\beta \in \mathcal{A}_l^d$.
Then, $\partial^\beta \in \mathcal{L}(\mathbb{P}_k^d, \mathbb{P}_k^d)$, i.e. $\partial^\beta$ is linear from $\mathbb{P}_k^d$ to $\mathbb{P}_k^d$, and for all $(a_\alpha)_{\alpha \in \mathcal{A}_k^d} \in \mathbb{R}$, we have

$$ (9.24) \quad \partial^\beta \left( \sum_{\alpha \in \mathcal{A}_k^d} a_\alpha \mathbf{X}^\alpha \right) = \sum_{\alpha \in \mathcal{A}_k^d} a_\alpha \partial^\beta \mathbf{X}^\alpha = \sum_{\substack{\alpha \in \mathcal{A}_k^d \\ \forall i \in [1..d], \beta_i \le \alpha_i}} a_\alpha \prod_{i=1}^d \left( \prod_{j=0}^{\beta_i-1} (\alpha_i - j) \right) X_i^{\alpha_i - \beta_i}. $$

*Proof.* Direct consequence of **the linearity of partial derivative**, Definition 1505 (polynomial space $\mathbb{P}_k^d$), and Lemma 1517 (partial derivative of monomials). □

**Lemma 1520 (partial derivative of 0).**
Let $d \ge 1$. Let $l \in \mathbb{N}$. Let $\beta \in \mathcal{A}_l^d$. Then, we have $\partial^\beta 0 = 0$.

*Proof.* Direct consequence of Lemma 1519 (partial derivative is linear). □

**Lemma 1521 (derivating more than degree is 0).**
Let $d \ge 1$. Let $k, l \in \mathbb{N}$. Let $\alpha \in \mathcal{C}_k^d$ and $\beta \in \mathcal{C}_l^d$. Assume that $k < l$. Then, we have $\partial^\beta \mathbf{X}^\alpha = 0$.

*Proof.* Case $\forall i \in [1..d]$, $\beta_i \le \alpha_i$. Then, from Definition 1486 (sets of multi-indices $\mathcal{A}_k^d$ and $\mathcal{C}_k^d$), and **the monotonicity of addition in $\mathbb{N}$**, we have $l = |\beta| \le |\alpha| = k$, which is impossible.

Case $\exists i \in [1..d]$, $\alpha_i < \beta_i$. Direct consequence of Lemma 1517 (partial derivative of monomials). □

**Lemma 1522 (partial derivative of monomials at 0).**
Let $d \ge 1$. Let $k, l \in \mathbb{N}$. Let $\alpha \in \mathcal{A}_k^d$ and $\beta \in \mathcal{A}_l^d$. Then, we have $\partial^\beta \mathbf{X}^\alpha(0) = \alpha! \delta_{\alpha,\beta}$.

*Proof.* From Definition 1482 (factorial of multi-indices), and Definition 1484 (Kronecker delta of multi-indices), we have $\alpha! \delta_{\alpha,\beta} = \prod_{i=1}^d \alpha_i!$ when $\alpha = \beta$, and 0 otherwise.

Case $\alpha = \beta$. Direct consequence of Lemma 1517 (partial derivative of monomials, since for all $i \in [1..d]$, $x_i^{\alpha_i - \beta_i} = 1$ and $\prod_{j=0}^{\beta_i - 1} (\alpha_i - j) = \alpha_i!$).

Case $\exists i \in [1..d]$, $\alpha_i \ne \beta_i$.

Case $\alpha_i < \beta_i$. Direct consequence of Lemma 1517 (partial derivative of monomials, $\partial^\beta \mathbf{X}^\alpha = 0$).

Case $\beta_i < \alpha_i$. Direct consequence of Lemma 1517 (partial derivative of monomials, since $x_i^{\alpha_i - \beta_i} = 0$ when $x_i \stackrel{\text{def.}}{=} 0$). □

**Lemma 1523 (monomials are free in $\mathbb{P}_k^d$).**
Let $d \ge 1$. Let $k \in \mathbb{N}$. Then, $(\mathbf{X}^\alpha)_{\alpha \in \mathcal{A}_k^d}$ is free in $\mathbb{P}_k^d$.

*Proof.* Let $(\lambda_\alpha)_{\alpha \in \mathcal{A}_k^d} \in \mathbb{R}$ such that $\sum_{\alpha \in \mathcal{A}_k^d} \lambda_\alpha \mathbf{X}^\alpha = 0$. Let $\beta \in \mathcal{A}_k^d$.
Then, from Lemma 1519 (partial derivative is linear), Lemma 1520 (partial derivative of 0), and Lemma 1522 (partial derivative of monomials at 0, with $\mathbf{x} \stackrel{\text{def.}}{=} \mathbf{0}$), we have

$$ 0 = \sum_{\alpha \in \mathcal{A}_k^d} \lambda_\alpha \partial^\beta \mathbf{X}^\alpha(0) = \sum_{\alpha \in \mathcal{A}_k^d} \lambda_\alpha \beta! \delta_{\alpha,\beta} = \lambda_\beta \beta!. $$

Thus, from Lemma 1483 (factorial of multi-index is positive), **the zero-product property in $\mathbb{R}$**, Definition 1505 (polynomial space $\mathbb{P}_k^d$, thus $\mathbf{X}^\alpha \in \mathbb{P}_k^d$), and **the definition of freedom**, we have $\lambda_\beta = 0$, and $(\mathbf{X}^\alpha)_{\alpha \in \mathcal{A}_k^d}$ is free in $\mathbb{P}_k^d$. □

**Lemma 1524 (monomials are a basis of $\mathbb{P}_k^d$).**
Let $d \ge 1$. Let $k \in \mathbb{N}$. Then, $(\mathbf{X}^\alpha)_{\alpha \in \mathcal{A}_k^d}$ is a basis of $\mathbb{P}_k^d$.
*Proof.* Direct consequence of Definition 1505 (*polynomial space* $\mathbb{P}_k^d$, thus monomials are generators), Lemma 1523 (*monomials are free in* $\mathbb{P}_k^d$), and **the definition of basis.** □

**Lemma 1525 (dimension of $\mathbb{P}_k^d$).**

Let $d \ge 1$. Let $k \in \mathbb{N}$. Then, $\mathbb{P}_k^d$ is a vector space of dimension

$$ (9.25) \qquad \dim \mathbb{P}_k^d = \binom{k+d}{d} = \binom{k+d}{k} = \frac{(k+d)!}{k! d!} \quad (= \mathrm{card}(\mathcal{A}_k^d)). $$

For instance, we have

$$ (9.26) \qquad \left\{ \begin{array}{l} \dim \mathbb{P}_0^d = 1, \\ \dim \mathbb{P}_1^d = d+1, \\ \dim \mathbb{P}_2^d = \frac{1}{2}(d+1)(d+2), \end{array} \right. \quad \text{and} \quad \left\{ \begin{array}{l} \dim \mathbb{P}_k^1 = (k+1), \\ \dim \mathbb{P}_k^2 = \frac{1}{2}(k+1)(k+2), \\ \dim \mathbb{P}_k^3 = \frac{1}{6}(k+1)(k+2)(k+3). \end{array} \right. $$

*Proof.* Direct consequence of Lemma 1524 (*monomials are a basis of* $\mathbb{P}_k^d$), **the definition of the dimension**, and Lemma 1498 (*cardinal of* $\mathcal{A}_k^d$). □

### 9.2.4 Decomposition of polynomials, isomorphisms of $\mathbb{P}_k^d$

**Lemma 1526 (isomorphism between $\mathbb{P}_0^d$ and $\mathbb{P}_0^{d-1}$).**

Let $d \ge 2$. Let $\xi^d \stackrel{\text{def.}}{=} ((\mathbf{x} \mapsto a_0) \in \mathbb{P}_0^d \longmapsto (\tilde{\mathbf{x}} \mapsto a_0) \in \mathbb{P}_0^{d-1})$.

Then, for all $p \in \mathbb{P}_0^d$, for all $\mathbf{x} \in \mathbb{R}^d$, we have $\xi^d(p)(\tilde{\mathbf{x}}) = p(\mathbf{x})$, and $\xi^d$ is an isomorphism.

*Proof.* Direct consequence of Lemma 1509 (*constant and affine spaces* $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$, thus $\xi^d$ is well-defined, and the identity holds), Lemma 1525 (*dimension of* $\mathbb{P}_k^d$, $\dim \mathbb{P}_0^d = \dim \mathbb{P}_0^{d-1} = 1$), **the definition of order (reflexivity)**, Lemma 1374 (*injectivity or surjectivity and dimension implies bijectivity*, $\xi^d$ is obviously linear and injective), and Definition 100 (*isomorphism*). □

**Remark 1527.**

Note in the next lemma the use of the tilde notation of Remark 1490, such that $\mathbf{x} = (\tilde{\mathbf{x}}, x_d)$.

Note also that the decomposition extracts $x_d$, and thus uses the horizontal slices $\tilde{S}_{k,i}^d$. A similar result, exhibiting $x_1$ and using vertical slices $\tilde{S}_{k,i}^d$, could be written, $p(\mathbf{x}) = \tilde{p}_0(\tilde{\mathbf{x}}) + x_1 p_1(\mathbf{x})$, with $\mathbf{x} = (x_1, \tilde{\mathbf{x}})$ and obvious modifications in the notations.

**Remark 1528.** See the sketch of the next proof in Section 3.2.

**Lemma 1529 (decomposition of $\mathbb{P}_k^d$).**

Then, there exist unique $\tilde{p}_0 \in \mathbb{P}_k^{d-1}$ and $p_1 \in \mathbb{P}_{k-1}^d$ such that

Let $d \ge 2$. Let $k \ge 1$. Let $p \in \mathbb{P}_k^d$.

$$ (9.27) \qquad \forall \mathbf{x} \in \mathbb{R}^d, \quad p(\mathbf{x}) = \tilde{p}_0(\tilde{\mathbf{x}}) + x_d p_1(\mathbf{x}) \quad (\text{i.e. } p = \tilde{p}_0 + X_d p_1), $$

which also writes $p(x_1, \dots, x_d) = \tilde{p}_0(x_1, \dots, x_{d-1}) + x_d p_1(x_1, \dots, x_d)$.

*Proof.* **Existence.** For all $k \ge 1$, let $P(k) \stackrel{\text{def.}}{=} [\forall p \in \mathbb{P}_k^d, \exists \tilde{p}_0 \in \mathbb{P}_k^{d-1}, \exists p_1 \in \mathbb{P}_{k-1}^d, p = \tilde{p}_0 + \mathbf{X}_d p_1]$.

**Induction: $P(1)$.** Let $p \in \mathbb{P}_1^d$. Then, from Lemma 1509 (*constant and affine spaces* $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$), and **commutative ring properties of $\mathbb{R}$** there exist

$$ a_0, (a_{\mathbf{e}_i})_{i \in [1..d]} \in \mathbb{R}, \quad \tilde{p}_0 \stackrel{\text{def.}}{=} a_0 + \sum_{i=1}^{d-1} a_{\mathbf{e}_i} X_i \in \mathbb{P}_1^{d-1}, \quad \text{and} \quad p_1 \stackrel{\text{def.}}{=} a_{\mathbf{e}_d} \in \mathbb{P}_0^d, $$

such that $p = a_0 + \sum_{i=1}^d a_{\mathbf{e}_i} X_i = \tilde{p}_0 + X_d p_1$.

**Induction: $P(k) \Rightarrow P(k+1)$.** Assume that $P(k)$ holds. Let $p \in \mathbb{P}_{k+1}^d$.
Then, from Definition 1505 (polynomial space $\mathbb{P}_k^d$), and Lemma 1496 ($C_k^d$ are layers of $\mathcal{A}_k^d$), there exist $(a_\alpha)_{\alpha \in \mathcal{A}_{k+1}^d} \in \mathbb{R}$, $q \stackrel{\text{def.}}{=} \sum_{\alpha \in \mathcal{A}_k^d} a_\alpha \mathbf{X}^\alpha \in \mathbb{P}_k^d$, and $r \stackrel{\text{def.}}{=} \sum_{\alpha \in \mathcal{C}_{k+1}^d} a_\alpha \mathbf{X}^\alpha \in \mathbb{P}_{k+1}^d$ such that $p = \sum_{\alpha \in \mathcal{A}_{k+1}^d} a_\alpha \mathbf{X}^\alpha = q+r$. By induction hypothesis, and from Lemma 1508 ($\mathbb{P}_k^d$ is nondecreasing sequence in $k$), there exist $\tilde{q}_0 \in \mathbb{P}_k^{d-1} \subset \mathbb{P}_{k+1}^{d-1}$ and $q_1 \in \mathbb{P}_{k-1}^d \subset \mathbb{P}_k^d$, such that $q = \tilde{q}_0 + X_d q_1$.

For all $i \in [0..k+1]$, for all $\tilde{\alpha}_i \in \mathcal{C}_{k+1-i}^{d-1}$, let $b_{\tilde{\alpha}_i} \stackrel{\text{def.}}{=} a_{(\tilde{\alpha}_i, i)} = a_{(\tilde{\varphi}_{k+1,d}^i)^{-1}(\tilde{\alpha}_i)}$. Then, from Lemma 1493 (slices of $\mathcal{C}_k^d$), Lemma 1494 (cardinal of slices of $\mathcal{C}_k^d$), and Definition 1491 (slices $\tilde{\mathcal{S}}_{k,i}^d$ and $\tilde{\mathcal{S}}_{k,i}^d$), we have

$$ 
\begin{aligned}
r &= \sum_{\alpha \in \tilde{\mathcal{S}}_{k+1,0}^d} a_\alpha \mathbf{X}^\alpha + \sum_{i=1}^{k+1} \sum_{\alpha \in \tilde{\mathcal{S}}_{k+1,i}^d} a_\alpha \mathbf{X}^\alpha \\
&= \underbrace{\sum_{\tilde{\alpha}_0 \in \mathcal{C}_{k+1}^{d-1}} b_{\tilde{\alpha}_0} \tilde{\mathbf{X}} \tilde{\alpha}_0}_{\stackrel{\text{def.}}{=} \tilde{r}_0} + X_d \underbrace{\sum_{i=1}^{k+1} \sum_{\tilde{\alpha}_i \in \mathcal{C}_{k+1-i}^{d-1}} b_{\tilde{\alpha}_i} \tilde{\mathbf{X}} \tilde{\alpha}_i X_d^{i-1}}_{\stackrel{\text{def.}}{=} r_1}.
\end{aligned}
$$

Let $i \in [1..k+1]$. Let $\tilde{\alpha}_i \in \mathcal{C}_{k+1-i}^{d-1}$. Then, from Lemma 1512 (monomials of $\mathcal{C}_k^d$ have degree $k$), we have $\text{deg}(\tilde{\mathbf{X}}\tilde{\alpha}_i X_d^{i-1}) = (k+1-i) + (i-1) = k$. Thus, from Lemma 1496 ($\mathcal{C}_k^d$ are layers of $\mathcal{A}_k^d$), and Definition 1505 (polynomial space $\mathbb{P}_k^d$), we have $\tilde{r}_0 \in \mathbb{P}_{k+1}^{d-1}$ and $r_1 \in \mathbb{P}_k^d$.

Let $\tilde{p}_0 \stackrel{\text{def.}}{=} \tilde{q}_0 + \tilde{r}_0$ and $p_1 \stackrel{\text{def.}}{=} q_1 + r_1$. Then, from ring properties of $\mathbb{R}$, and Lemma 1507 ($\mathbb{P}_k^d$ is vector space), we have $\tilde{p}_0 \in \mathbb{P}_{k+1}^{d-1}$, $p_1 \in \mathbb{P}_k^d$, and $p = \tilde{p}_0 + X_d p_1$, i.e. $P(k+1)$ holds. This concludes the induction on $k$, and we have existence for all $k \ge 1$.

**Uniqueness.** Let $p \in \mathbb{P}_k^d$, $\tilde{q}_0, \tilde{r}_0 \in \mathbb{P}_k^{d-1}$, and $q_1, r_1 \in \mathbb{P}_{k-1}^d$, such that,

$$ \forall \mathbf{x} \in \mathbb{R}^d, \quad p(\mathbf{x}) = \tilde{q}_0(\tilde{\mathbf{x}}) + x_d q_1(\mathbf{x}) = \tilde{r}_0(\tilde{\mathbf{x}}) + x_d r_1(\mathbf{x}). $$

Let $\tilde{\mathbf{x}} \in \mathbb{R}^{d-1}$. Let $\mathbf{x} \stackrel{\text{def.}}{=} (\tilde{\mathbf{x}}, 0)$. Then, we have $\tilde{q}_0(\tilde{\mathbf{x}}) = \tilde{r}_0(\tilde{\mathbf{x}})$, i.e. $\tilde{q}_0 = \tilde{r}_0$, thus $X_d(q_1 - r_1) = 0$. Thus, from Lemma 1507 ($\mathbb{P}_k^d$ is vector space), Definition 1505 (polynomial space $\mathbb{P}_k^d$), Lemma 1523 (monomials are free in $\mathbb{P}_k^d$), and the definition of freedom, we have $q_1 - r_1 \in \mathbb{P}_{k-1}^d$ and there exists $(a_\alpha)_{\alpha \in \mathcal{A}_{k-1}^d} \in \mathbb{R}$ such that $q_1 - r_1 = \sum_{\alpha \in \mathcal{A}_{k-1}^d} a_\alpha \mathbf{X}^\alpha$, and thus

$$ X_d(q_1 - r_1) = \sum_{\alpha \in \mathcal{A}_{k-1}^d} a_\alpha \mathbf{X}^{\alpha+e_d} = 0 \quad (\text{in } \mathbb{P}_k^d), $$

which implies $a_\alpha = 0$ for all $\alpha \in \mathcal{A}_{k-1}^d$, i.e. $q_1 = r_1$.

Therefore, we have uniqueness. $\square$

**Remark 1530.** Lemmas 1529 and 1531 allow to write $\dim(\mathbb{P}_k^d) = \dim(\mathbb{P}_k^{d-1})+\dim(\mathbb{P}_{k-1}^d)$. Actually, this could be a way to prove (by induction) that $\dim(\mathbb{P}_k^d) = \binom{k+d}{d}$. But, we preferred to prove directly that $\text{card}(\mathcal{C}_k^d) = \binom{k+d-1}{d-1}$, by working on slices of $\mathcal{C}_k^d$.

**Lemma 1531 (isomorphism between $\mathbb{P}_k^d$ and $\mathbb{P}_k^{d-1} \times \mathbb{P}_{k-1}^d$).**

Let $d \ge 2$. Let $k \ge 1$. Let $\zeta_k^d \stackrel{\text{def.}}{=} (p \in \mathbb{P}_k^d \longmapsto (\tilde{p}_0, p_1) \in \mathbb{P}_k^{d-1} \times \mathbb{P}_{k-1}^d)$ with $p = \tilde{p}_0 + X_d p_1$. Then, $\zeta_k^d$ is an isomorphism.

**Proof.** From Lemma 1529 (decomposition of $\mathbb{P}_k^d$), $\zeta_k^d$ is well-defined for all $p \in \mathbb{P}_k^d$.

**Linearity.** Let $p, q \in \mathbb{P}_k^d$, and $\lambda, \mu \in \mathbb{R}$. Then, from Lemma 1529 (decomposition of $\mathbb{P}_k^d$), commutative ring properties of $\mathbb{R}$, Lemma 1507 ($\mathbb{P}_k^d$ is vector space), and Lemma 82 (closed under
linear combination is subspace), there exist unique $\tilde{p}_0, \tilde{q}_0 \in \mathbb{P}_k^{d-1}$ and $p_1, q_1 \in \mathbb{P}_{k-1}^d$ such that $p = \tilde{p}_0 + X_d p_1$ and $q = \tilde{q}_0 + X_d q_1$, thus $\lambda p + \mu q = \lambda \tilde{p}_0 + \mu \tilde{q}_0 + X_d (\lambda p_1 + \mu q_1)$ with $\lambda \tilde{p}_0 + \mu \tilde{q}_0 \in \mathbb{P}_k^{d-1}$ and $\lambda p_1 + \mu q_1 \in \mathbb{P}_{k-1}^d$. Hence, from Lemma 1529 (decomposition of $\mathbb{P}_k^d$, for $\lambda p + \mu q$, uniqueness), Definition 89 (product vector operations), Lemma 90 (product is space), and Lemma 95 (linear map preserves linear combinations), we have

$$ \zeta_k^d(\lambda p + \mu q) = (\lambda \tilde{p}_0 + \mu \tilde{q}_0, \lambda p_1 + \mu q_1) = \lambda(\tilde{p}_0, p_1) + \mu(\tilde{q}_0, q_1) = \lambda \zeta_k^d(p) + \mu \zeta_k^d(q), $$

and $\zeta_k^d$ is a linear map from $\mathbb{P}_k^d$ to the product vector space $\mathbb{P}_k^{d-1} \times \mathbb{P}_{k-1}^d$.

**Injectivity.** Let $p \in \mathbb{P}_k^d$. Assume that $\zeta_k^d(p) = 0$. Then, from Lemma 1529 (decomposition of $\mathbb{P}_k^d$), Definition 101 (kernel), and Lemma 103 (injective linear map has zero kernel), there exist unique $\tilde{p}_0 \in \mathbb{P}_k^{d-1}$ and $p_1 \in \mathbb{P}_{k-1}^d$ such that $p = \tilde{p}_0 + X_d p_1$, thus $\zeta_k^d(p) = (\tilde{p}_0, p_1) = (0, 0) = (0, 0)$, and $\zeta_k^d$ is injective.

**Dimension.** From Lemma 1525 (dimension of $\mathbb{P}_k^d$, three times), Lemma 1364 (properties of the binomial coefficient, (5.7)), and the rule of dimension for product of spaces, we have

$$ \dim(\mathbb{P}_k^d) = \binom{k+d}{d} = \binom{k+d-1}{d-1} + \binom{k-1+d}{d} = \dim(\mathbb{P}_k^{d-1}) + \dim(\mathbb{P}_{k-1}^d) = \dim(\mathbb{P}_k^{d-1} \times \mathbb{P}_{k-1}^d). $$

**Isomorphism.** Direct consequence of the definition of order (reflexivity), Lemma 1374 (injectivity or surjectivity and dimension implies bijectivity), and Definition 100 (isomorphism). □

**Remark 1532.** Note in the next lemma that, by an abuse of notation, for all $p \in \mathbb{P}_k^d$, the function that sends $(\mathbf{x}, x_{d+1})$ to $p(\mathbf{x})$ is sometimes still denoted $p$.

**Lemma 1533 (P<sub>k</sub><sup>d</sup> is nondecreasing sequence in d).**
Let $k \in \mathbb{N}$.
Then, the sequence $(\mathbb{P}_k^d)_{d \ge 1}$ is nondecreasing for the inclusion, in the sense that, for all $d \ge 1$, for all $p \in \mathbb{P}_k^d$, the function $((\mathbf{x}, x_{d+1}) \in \mathbb{R}^d \times \mathbb{R} \mapsto p(\mathbf{x}) \in \mathbb{R})$ belongs to $\mathbb{P}_k^{d+1}$.

**Proof.** Let $d \ge 1$. Let $p \in \mathbb{P}_k^d$.

**Case k = 0.** Direct consequence of Lemma 1509 (constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$).

**Case k $\ge 1$.** Let $\tilde{p}_0 \stackrel{\text{def.}}{=} p \in \mathbb{P}_k^d$ and $p_1 \stackrel{\text{def.}}{=} 0$. Let $\mathbf{x} \in \mathbb{R}^d$ and $x_{d+1} \in \mathbb{R}$. Then, from Lemma 1507 ($\mathbb{P}_k^d$ is vector space, thus $p_1 = 0 \in \mathbb{P}_{k-1}^{d+1}$), Lemma 1531 (isomorphism between $\mathbb{P}_k^d$ and $\mathbb{P}_k^{d-1} \times \mathbb{P}_{k-1}^d$), and Definition 1505 (polynomial space $\mathbb{P}_k^d$), we have $\hat{p} = (\zeta_k^{d+1})^{-1}(\tilde{p}_0, p_1) \in \mathbb{P}_k^{d+1}$, and $\hat{p}(\mathbf{x}, x_{d+1}) = p(\mathbf{x}) + x_{d+1} = p(\mathbf{x})$. □

**Lemma 1534 (expression of $\mathbb{P}_k^d$ as polynomial of $x_d$).** Let $d \ge 2$. Let $k \in \mathbb{N}$. Let $p \in \mathbb{P}_k^d$. Then, for all $i \in [0..k]$, there exists a unique $\tilde{r}_i \in \mathbb{P}_{k-i}^{d-1}$, such that

$$ (9.28) \qquad \forall \mathbf{x} \in \mathbb{R}^d, \quad p(\mathbf{x}) = \sum_{i=0}^{k} \tilde{r}_i(\tilde{\mathbf{x}})x_d^i \qquad \left( \text{i.e. } p = \sum_{i=0}^{k} \tilde{r}_i X_d^i \right). $$

Thus, for all $\tilde{\mathbf{x}} \in \mathbb{R}^{d-1}$, the function $(x_d \mapsto p(\tilde{\mathbf{x}}, x_d))$ belongs to $\mathbb{P}_k^1$.

**Proof. Existence.** For all $k \in \mathbb{N}$, let $P(k) \stackrel{\text{def.}}{=} [\forall p \in \mathbb{P}_k^d, (\forall i \in [0..k], \exists \tilde{r}_i \in \mathbb{P}_{k-i}^{d-1}), p = \sum_{i=0}^{k} \tilde{r}_i X_d^i]$.

**Induction: $P(0)$.** Let $p \in \mathbb{P}_0^d$. Then, from Lemma 1526 (isomorphism between $\mathbb{P}_0^d$ and $\mathbb{P}_0^{d-1}$), $\tilde{r}_0 \stackrel{\text{def.}}{=} \xi^d(p) \in \mathbb{P}_0^{d-1}$ is such that, for all $\tilde{\mathbf{x}} \in \mathbb{R}^{d-1}$, for all $x_d \in \mathbb{R}$, $p(\tilde{\mathbf{x}}, x_d) = \tilde{r}_0(\tilde{\mathbf{x}})$, i.e. $P(0)$ holds.

**Induction: $P(1)$.** Let $p \in \mathbb{P}_1^d$. Then, from Lemma 1529 (decomposition of $\mathbb{P}_k^d$), there exists $\tilde{r}_0 \in \mathbb{P}_1^{d-1}$ and $q_1 \in \mathbb{P}_0^d$, such that $p = \tilde{r}_0 + X_d q_1$. Thus, from Lemma 1526 (isomorphism between
$\mathbb{P}_0^d$ and $\mathbb{P}_0^{d-1}$), $\tilde{r}_1 \stackrel{\text{def.}}{=} \xi^d(q_1) \in \mathbb{P}_0^{d-1}$ is such that, for all $\tilde{\mathbf{x}} \in \mathbb{R}^{d-1}$, for all $x_d \in \mathbb{R}$, $q_1(\tilde{\mathbf{x}}, x_d) = \tilde{r}_1(\tilde{\mathbf{x}})$, hence $p = \tilde{r}_0 + X_d\tilde{r}_1$ with $\tilde{r}_0 \in \mathbb{P}_1^{d-1}$ and $\tilde{r}_1 \in \mathbb{P}_0^{d-1}$, i.e. $P(1)$ holds.

**Induction:** $P(k) \Rightarrow P(k+1)$. Let $k \ge 1$. Assume that $P(k)$ holds. Let $p \in \mathbb{P}_{k+1}^d$. Then, from Lemma 1529 (decomposition of $\mathbb{P}_k^d$), there exist $\tilde{r}_0 \in \mathbb{P}_{k+1}^{d-1}$ and $q_1 \in \mathbb{P}_k^d$, such that $p = \tilde{r}_0 + X_d q_1$. By induction hypothesis, for all $i \in [0..k]$, there exist $\tilde{s}_i \in \mathbb{P}_{k-i}^{d-1}$, such that $q_1 = \sum_{i=0}^k \tilde{s}_i X_d^i$. For all $i \in [1..k+1]$, let $\tilde{r}_i \stackrel{\text{def.}}{=} \tilde{s}_{i-1} \in \mathbb{P}_{k+1-i}^{d-1}$. Then, from commutative ring properties of $\mathbb{R}$, we have $p = \sum_{i=0}^{k+1} \tilde{r}_i X_d^i$, i.e. $P(k+1)$ holds.

This concludes the induction on $k$, and we have existence for all $k \in \mathbb{N}$.

**Uniqueness.** Let $p \in \mathbb{P}_k^d$, and for all $i \in [0..k]$, let $\tilde{r}_i, \tilde{s}_i \in \mathbb{P}_{k-i}^{d-1}$ such that

$$p = \sum_{i=0}^{k} \tilde{r}_i X_d^i = \sum_{i=0}^{k} \tilde{s}_i X_d^i.$$

Let $\tilde{\mathbf{x}} \in \mathbb{R}^{d-1}$. Then, from ring properties of $\mathbb{R}$, and Lemma 1419 (monomials are free in $\mathbb{P}_k^1$), we have $\sum_{i=0}^k (\tilde{r}_i(\tilde{\mathbf{x}}) - \tilde{s}_i(\tilde{\mathbf{x}}))X_d^i = 0$, and thus, for all $i \in [0..k]$, $\tilde{r}_i(\tilde{\mathbf{x}}) - \tilde{s}_i(\tilde{\mathbf{x}}) = 0$, i.e. $\tilde{r}_i = \tilde{s}_i$.

Therefore, we have uniqueness. $\square$

*Remark* 1535. Lemma 1534 expresses a *d*-multivariate polynomial as an univariate polynomial with coefficients that are (*d* − 1)-multivariate polynomials. The proof uses Horner's rule,

$$p = \tilde{r}_0 + X_d(\tilde{r}_1 + X_d(\tilde{r}_2 + X_d(\dots + X_d(\tilde{r}_{k-1} + X_d\tilde{r}_k)\dots))).$$

### 9.2.5 Product of polynomials (alternate)

*Remark* 1536. The next lemma provides an alternate proof for Lemma 1516. This second proof avoids the double sum on multi-indices that is not explicitly formulated in the first proof. It relies on the decomposition of polynomials of Lemma 1529 and on a strong double induction.

**Lemma 1537 (product of two polynomials (alternate proof)).**

Let $d \ge 1$. Let $k, l \in \mathbb{N}$. Let $p \in \mathbb{P}_k^d$ and $q \in \mathbb{P}_l^d$. Then, we have $pq \in \mathbb{P}_{k+l}^d$.

*Proof.* For all $d \ge 1$, for all $n \in \mathbb{N}$, let $P$ be the property defined by

$$P(d, n) \stackrel{\text{def.}}{=} [\forall k, l \in \mathbb{N}, \forall p \in \mathbb{P}_k^d, \forall q \in \mathbb{P}_l^d, k+l \le n \Rightarrow pq \in \mathbb{P}_n^d].$$

Then, from Lemma 1508 ($\mathbb{P}_k^d$ is nondecreasing sequence in $k$, thus $k+l \le n \Rightarrow \mathbb{P}_{k+l}^d \subseteq \mathbb{P}_n^d$), the result to prove is equivalent to establish $P$. Let $d \ge 1$ and $n \in \mathbb{N}$.

**Strong double induction.**

Assume that for all $d_1, n_1 \in \mathbb{N}$, $1 \le d_1 \le d$, $n_1 \le n$, and $(d_1, n_1) \ne (d, n)$ implies $P(d_1, n_1)$. Let us show $P(d, n)$. Let $k, l \in \mathbb{N}$. Let $p \in \mathbb{P}_k^d$ and $q \in \mathbb{P}_l^d$. Assume that $k+l \le n$. Let us show $pq \in \mathbb{P}_n^d$.

**Case $d = 1$.** Direct consequence of Lemma 1422 (product of two univariate polynomials).

**Case $k = 0$ or $l = 0$.** Direct consequence of Lemma 1509 (constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$, thus $p = a_0\mathbf{X}^0$ (resp. $q = b_0\mathbf{X}^0$) is constant), commutative ring properties of $\mathbb{R}$ (thus $pq = a_0q$ (resp. $pq = b_0p$)), Lemma 1507 ($\mathbb{P}_k^d$ is vector space), Lemma 81 (closed under vector operations is subspace, closed under scalar multiplication, thus $pq$ is in $\mathbb{P}_{k+l}^d$), and Lemma 1508 ($\mathbb{P}_k^d$ is nondecreasing sequence in $k$, thus $\mathbb{P}_{k+l}^d \subseteq \mathbb{P}_n^d$).

**Case $k+l < n$.** Then, by hypothesis, $P(d, k+l)$ holds, i.e. $pq \in \mathbb{P}_n^d$.

**Case $d \ge 2$, $k \ne 0$, $l \ne 0$ and $k+l = n \ge 1$.**

Then, from Lemma 1529 (decomposition of $\mathbb{P}_k^d$), and commutative ring properties of $\mathbb{R}$, there exist $\tilde{p}_0 \in \mathbb{P}_k^{d-1}$, $p_1 \in \mathbb{P}_{k-1}^d$, $\tilde{q}_0 \in \mathbb{P}_l^{d-1}$ and $q_1 \in \mathbb{P}_{l-1}^d$, such that

$$p = \tilde{p}_0 + X_d p_1, \quad q = \tilde{q}_0 + X_d q_1, \quad \text{thus} \quad pq = \tilde{p}_0 \tilde{q}_0 + X_d (\tilde{p}_0 q_1 + p_1 \tilde{q}_0) + X_d^2 p_1 q_1.$$
Then, by hypothesis (with $d_1 \stackrel{\text{def.}}{=} d - 1 < d$ and $n_1 \stackrel{\text{def.}}{=} n$), and from Lemma 1533 ($\mathbb{P}_k^d$ is nondecreasing sequence in $d$), $P(d-1, n)$ holds, i.e. $\tilde{p}_0\tilde{q}_0 \in \mathbb{P}_n^{d-1}$, and thus

$$ (\mathbf{x} \stackrel{\text{def.}}{=} (\tilde{\mathbf{x}}, x_d) \in \mathbb{R}^d \longmapsto (\tilde{p}_0\tilde{q}_0)(\tilde{\mathbf{x}})) \in \mathbb{P}_n^d. $$

Next, from Lemma 1533 ($\mathbb{P}_k^d$ is nondecreasing sequence in $d$), and by hypothesis (with $d_1 \stackrel{\text{def.}}{=} d$ and $n_1 \stackrel{\text{def.}}{=} n - 1 < n$), we have $(\mathbf{x} \mapsto \tilde{p}_0(\tilde{\mathbf{x}})) \in \mathbb{P}_k^d$, $(\mathbf{x} \mapsto \tilde{q}_0(\tilde{\mathbf{x}})) \in \mathbb{P}_l^d$, and $P(d, n-1)$ holds, i.e. $(\mathbf{x} \mapsto \tilde{p}_0(\tilde{\mathbf{x}})q_1(\mathbf{x}))$ and $(\mathbf{x} \mapsto p_1(\mathbf{x})\tilde{q}_0(\tilde{\mathbf{x}}))$ belong to $\mathbb{P}_{n-1}^d$. Hence, from Lemma 1507 ($\mathbb{P}_k^d$ is vector space), Lemma 81 (closed under vector operations is subspace), and Lemma 1515 (product of monomial and polynomial, with $\alpha \stackrel{\text{def.}}{=} e_d$), we have $\tilde{p}_0q_1 + p_1\tilde{q}_0 \in \mathbb{P}_{n-1}^d$, and $X_d(\tilde{p}_0q_1 + p_1\tilde{q}_0) \in \mathbb{P}_n^d$. Moreover, by hypothesis (with $d_1 \stackrel{\text{def.}}{=} d$ and $n_1 \stackrel{\text{def.}}{=} n - 2 < n$), and from Lemma 1515 (product of monomial and polynomial, with $\alpha \stackrel{\text{def.}}{=} 2e_d$), $P(d, n-2)$ holds, i.e. $p_1q_1 \in \mathbb{P}_{n-2}^d$, and $X_d^2 p_1q_1 \in \mathbb{P}_n^d$. Finally, from Lemma 1507 ($\mathbb{P}_k^d$ is vector space), and Lemma 81 (closed under vector operations is subspace), we have $pq \in \mathbb{P}_n^d$.

Therefore, we always have $pq \in \mathbb{P}_n^d$, and from Lemma 1360 (strong double induction), the property holds for all $d \ge 1$ and for all $n \in \mathbb{N}$. $\square$

**Lemma 1538 (product of polynomials).** Let $d, n \ge 1$. Let $(k_i)_{i \in [1..n]} \in \mathbb{N}$.

Let $k \stackrel{\text{def.}}{=} \sum_{i=1}^n k_i$. For all $i \in [1..n]$, let $p_i \in \mathbb{P}_{k_i}^d$. Then, we have $\prod_{i=1}^n p_i \in \mathbb{P}_k^d$.

*Proof.* Direct consequence (by induction on $n$) of Lemma 1516 (product of two polynomials), or Lemma 1537 (product of two polynomials (alternate proof)). $\square$

### 9.2.6 Composition of polynomials and affine mappings

**Lemma 1539 (affine mapping of monomials is $\mathbb{P}_k^l$).**

Let $l, d \ge 1$. Let $k \in \mathbb{N}$. Let $f \in \mathcal{A}ff(\mathbb{R}^l, \mathbb{R}^d)$. Let $\alpha \in \mathcal{C}_k^d$. Then, we have $\mathbf{X}^\alpha \circ f \in \mathbb{P}_k^l$.

*Proof.* Let $q \stackrel{\text{def.}}{=} \mathbf{X}^\alpha \circ f$.

For all $i \in [1..d]$, let $f_i$ be the $i$-th component of $f$. Let $\mathbf{y} \in \mathbb{R}^l$. Then, from Lemma 1399 (equivalent definition of affine map (finite dimension)), Lemma 1509 (constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$), and Definition 1503 (monomial in $d$ variables), $f_i$ belongs to $\mathbb{P}_1^l$, and we have

$$ q(\mathbf{y}) = \prod_{i=1}^{d} [f_i(\mathbf{y})]^{\alpha_i}, \quad \text{and} \quad [f_i(\mathbf{y})]^{\alpha_i} = \prod_{j=1}^{\alpha_i} f_j(\mathbf{y}). $$

Thus, from Lemma 1538 (product of polynomials, with $n \stackrel{\text{def.}}{=} \alpha_i$ and $p_j \stackrel{\text{def.}}{=} f_i \in \mathbb{P}_1^l$ for all $j \in [1..\alpha_i]$, then with $n \stackrel{\text{def.}}{=} d$ and $p_i \stackrel{\text{def.}}{=} f_i^{\alpha_i} \in \mathbb{P}_{\alpha_i}^l$ for all $i \in [1..d]$, thus $q \in \mathbb{P}_{\sum_{i=1}^d \alpha_i}^l$), Definition 1486 (sets of multi-indices $\mathcal{A}_k^d$ and $\mathcal{C}_k^d$, thus $|\alpha| = k$), and Definition 1480 (length of multi-indices), we have $q \in \mathbb{P}_k^l$. $\square$

**Lemma 1540 (affine mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$).**

Let $l, d \ge 1$. Let $k \in \mathbb{N}$. Let $f \in \mathcal{A}ff(\mathbb{R}^l, \mathbb{R}^d)$. Let $p \in \mathbb{P}_k^d$. Then, we have $p \circ f \in \mathbb{P}_k^l$.

*Proof.* From Definition 1505 (polynomial space $\mathbb{P}_k^d$), distributivity of composition over addition, Lemma 1496 ($\mathcal{C}_k^d$ are layers of $\mathcal{A}_k^d$), Lemma 1539 (affine mapping of monomials is $\mathbb{P}_k^l$, thus $\mathbf{X}^\alpha \circ f \in \mathbb{P}_i^l$ for all $i \in [0..k]$ and for all $\alpha \in \mathcal{C}_i^d$), and Lemma 1508 ($\mathbb{P}_k^d$ is nondecreasing sequence in $k$, thus $\mathbb{P}_i^l \subset \mathbb{P}_k^l$ for all $i \in [0..k]$), and Lemma 1507 ($\mathbb{P}_k^d$ is vector space), we have

$$ q \stackrel{\text{def.}}{=} p \circ f = \left( \sum_{\alpha \in \mathcal{A}_k^d} a_\alpha \mathbf{X}^\alpha \right) \circ f = \sum_{\alpha \in \mathcal{A}_k^d} a_\alpha (\mathbf{X}^\alpha \circ f) = \sum_{i=0}^{k} \sum_{\alpha \in \mathcal{C}_i^d} a_\alpha (\mathbf{X}^\alpha \circ f), $$

and $q \in \mathbb{P}_k^l$. $\square$
## 9.3 $\mathbb{P}_1^d$ affine polynomials and affine geometric mapping

### 9.3.1 Reference affine Lagrange polynomials

**Definition 1541 (reference Lagrange polynomials of $\mathbb{P}_1^d$).** Let $d \ge 1$. Let $i \in [0..d]$. Let $\hat{x} \in \mathbb{R}^d$. The $i$-th reference Lagrange polynomials of $\mathbb{P}_1^d$ is denoted $\hat{\mathcal{L}}_i^{1,d}$, and is defined by

$$ (9.29) \qquad (i = 0) \qquad \hat{\mathcal{L}}_0^{1,d}(\hat{x}) \stackrel{\text{def.}}{=} 1 - \sum_{j=1}^{d} \hat{x}_j, $$

$$ (9.30) \qquad (i \in [1..d]) \qquad \hat{\mathcal{L}}_i^{1,d}(\hat{x}) \stackrel{\text{def.}}{=} \hat{x}_i. $$

**Lemma 1542 (reference Lagrange polynomials of $\mathbb{P}_1^d$ for $d = 1$ are reference Lagrange polynomials of $\mathbb{P}_k^1$ for $k = 1$).**

The reference Lagrange polynomials of $\mathbb{P}_1^d$ for $d \stackrel{\text{def.}}{=} 1$ and the reference Lagrange polynomials of $\mathbb{P}_k^1$ for $k \stackrel{\text{def.}}{=} 1$ from Lemma 1455 (reference Lagrange basis of $\mathbb{P}_k^1$) coincide.

*Proof.* Direct consequence of Definition 1541 (reference Lagrange polynomials of $\mathbb{P}_1^d$, with $d \stackrel{\text{def.}}{=} 1$), Definition 1448 (Lagrange polynomials of $\mathbb{P}_k^1$, with $k \stackrel{\text{def.}}{=} 1$), Definition 1453 (reference Lagrange nodes of $\mathbb{P}_k^1$, (8.9) with $k \stackrel{\text{def.}}{=} 1$), Lemma 1455 (reference Lagrange basis of $\mathbb{P}_k^1$, with $k \stackrel{\text{def.}}{=} 1$), and field properties of $\mathbb{R}$ (with $1 - 0 \ne 0$ and $0 - 1 \ne 0$ for $k \stackrel{\text{def.}}{=} 1$). $\square$

**Lemma 1543 (reference Lagrange polynomials is basis of $\mathbb{P}_1^d$).** Let $d \ge 1$. Then, the reference Lagrange polynomials $(\hat{\mathcal{L}}_i^{1,d})_{i \in [0..d]}$ form a basis of $\mathbb{P}_1^d$, that satisfies

$$ (9.31) \qquad \forall i \in [0..d], \quad \deg \hat{\mathcal{L}}_i^{1,d} = 1, $$

$$ (9.32) \qquad \forall i, j \in [0..d], \quad \hat{\mathcal{L}}_i^{1,d}(\hat{v}_j) = \delta_{ij}, $$

$$ (9.33) \qquad \sum_{i=0}^{d} \hat{\mathcal{L}}_i^{1,d} = 1. $$

*Proof. Degree (9.31), in $\mathbb{P}_1^d$, and identities (9.32) and (9.33).*

Direct consequence of Definition 1541 (reference Lagrange polynomials of $\mathbb{P}_1^d$), Lemma 1509 (constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$), Definition 1510 (degree of polynomial), additive group properties of $\mathbb{R}$, and Definition 1434 (family of reference points).

**Basis.** Let $(\lambda_i)_{i \in [0..d]} \in \mathbb{R}$ such that, for all $\hat{x} \in \mathbb{R}^d$, $\sum_{i=0}^d \lambda_i \hat{\mathcal{L}}_i^{1,d}(\hat{x}) = 0$. Then, from (9.32), and ring properties of $\mathbb{R}$, taking $\hat{x} \stackrel{\text{def.}}{=} \hat{v}_j$ provides $\sum_{i=0}^d \lambda_i \delta_{ij} = \lambda_j = 0$. Thus, from the definition of freedom, Lemma 1525 (dimension of $\mathbb{P}_k^d$, with $k \stackrel{\text{def.}}{=} 1$), and Lemma 1376 (free family of dim elements is basis), the reference Lagrange polynomials form a basis of $\mathbb{P}_1^d$. $\square$

**Remark 1544.** In the next lemma, the linear maps from $\mathbb{R}^n$ to $\mathbb{R}^p$ are assimilated to their matrix relative to the canonical bases.

**Lemma 1545 (differential of reference Lagrange polynomials).** Let $d \ge 1$. Let $\hat{x} \in \mathbb{R}^d$. Let $i \in [0..d]$. Then, we have $\hat{\mathcal{L}}_i^{1,d} \in C^\infty(\mathbb{R}^d, \mathbb{R})$ and

$$ (9.34) \qquad (i = 0) \qquad \mathrm{D} \hat{\mathcal{L}}_0^{1,d}(\hat{x}) = -\mathbf{1} \in \mathcal{M}_{1,d}(\mathbb{R}), $$

$$ (9.35) \qquad (i \in [1..d]) \qquad \mathrm{D} \hat{\mathcal{L}}_i^{1,d}(\hat{x}) = \mathbf{e}_i \in \mathcal{M}_{1,d}(\mathbb{R}). $$
*Proof.* Direct consequence of Lemma 1543 (*reference Lagrange polynomials is basis of $\mathbb{F}_1^d$*), Lemma 1509 (*constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$, thus reference Lagrange polynomials are affine maps*), *affine maps in $\mathbb{R}^d$ are $\mathcal{C}^\infty$, and the rules of derivation (thus affine maps have constant differential and the differential is the linear part).* □

### 9.3.2 Affine geometric mapping from $\mathbb{R}^d$ to $\mathbb{R}^d$

*Remark 1546.* In Figure 9.4, we plot an example of $\varphi_{\text{geo}}^{[\mathbf{v}]^d}$, the affine geometric mapping from the reference simplex in $\mathbb{R}^d$ to the current simplex in $\mathbb{R}^d$. This mapping transforms the *i*-th hyperface (i.e. opposite the vertex $i \in [0..d]$) of the reference simplex to the *i*-th hyperface of the current simplex, see Figure 9.5.

Other affine geometric mappings are also introduced. First, in Definition 1578, the affine geometric mapping $\varphi_{\pi^l}^{[\mathbf{v}]^d}: \mathbb{R}^l \to \mathbb{R}^d$ (for $l \in [1..d]$) is a generalization of $\varphi_{\text{geo}}^{[\mathbf{v}]^d}$, that allows to pass from the reference *l*-simplex to the current *d*-simplex. The injective mapping $\pi^l: [0..l] \to [0..d]$ is used to select the chosen *l*-face of the *d*-simplex.

Next, one can define the geometric mapping $\varphi_{\theta_i^{d-1}}^{[\mathbf{v}]^d}: \mathbb{R}^{d-1} \to \mathbb{R}$ (for $i \in [0..d]$), see Lemma 1584. It transfers the reference ($d-1$)-simplex to the *i*-th hyperface of the current *d*-simplex, see Figure 9.6.

Finally, the geometric mapping $\varphi_{\pi^d}^{[\mathbf{v}]^d}: \mathbb{R}^d \to \mathbb{R}^d$ is defined in Lemma 1586. It transforms the reference *d*-simplex to a “permutation” of the current *d*-simplex (i.e. with a permutation of the vertices), see Figure 9.7 in the case of a circular permutation (defined in Lemma 1366). It is used in the proof of the factorization of polynomial Lemma 1623.

**Definition 1547 (geometric mapping).** Let $d \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ points in $\mathbb{R}^d$. The geometric mapping associated with $[\mathbf{v}]^d$ is denoted $\varphi_{\text{geo}}^{[\mathbf{v}]^d}$, and is defined by

$$ (9.36) \qquad \forall \hat{\mathbf{x}} \in \mathbb{R}^d, \quad \varphi_{\text{geo}}^{[\mathbf{v}]^d}(\hat{\mathbf{x}}) = \sum_{i=0}^{d} \hat{\mathcal{L}}_{i}^{1,d}(\hat{\mathbf{x}}) \mathbf{v}_i \quad \in \mathbb{R}^d. $$

**Lemma 1548 (geometric mapping for $d=1$ is geometric mapping in dimension 1).**
Let $[\mathbf{v}]^1 = [\mathbf{v}]^1$ be two points in $\mathbb{R}^1$.

Then, $\varphi_{\text{geo}}^{[\mathbf{v}]^d}$ for $d \stackrel{\text{def.}}{=} 1$ and $\varphi_{\text{geo}}^{[\mathbf{v}]^1}$ from Definition 1461 coincide.

*Proof.* Direct consequence of Definition 1547 (*geometric mapping*, with $d \stackrel{\text{def.}}{=} 1$), Definition 1541 (*reference Lagrange polynomials of $\mathbb{P}_1^d$, with $d \stackrel{\text{def.}}{=} 1$*), Definition 1461 (*geometric mapping in dimension 1*), and **commutative ring properties of $\mathbb{R}$**. □

**Lemma 1549 (reference geometric mapping is identity).**
Let $d \ge 1$. Then, the geometric mapping for reference vertices $[\hat{\mathbf{v}}]^d$ is the identity, $\varphi_{\text{geo}}^{[\hat{\mathbf{v}}]^d} = \text{Id}_{\mathbb{R}^d}$.

*Proof.* Direct consequence of Definition 1547 (*geometric mapping*), Definition 1541 (*reference Lagrange polynomials of $\mathbb{P}_1^d$*), and Definition 1434 (*family of reference points*). □
<figure><img src="image_22.png" /><figcaption>Figure 9.4: Geometric mapping $\varphi_{\text{geo}}^{[\mathbf{v}]^d}$ in the case $d = 2$, see Definition 1547.</figcaption></figure>

**Lemma 1550 (properties of geometric mapping).**
Let $d \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ points in $\mathbb{R}^d$. Let $A_{\text{geo}}^{[\mathbf{v}]^d} \stackrel{\text{def.}}{=} (\mathbf{v}_1 - \mathbf{v}_0 \ \mathbf{v}_2 - \mathbf{v}_0 \ \dots \ \mathbf{v}_d - \mathbf{v}_0)$ in $\mathcal{M}_{d,d}$.

Then, $\varphi_{\text{geo}}^{[\mathbf{v}]^d}$ belongs to $\mathcal{A}\mathcal{F}(\mathbb{R}^d, \mathbb{R}^d)$, and we have

$$ (9.37) \quad \forall j \in [0..d], \quad \forall \hat{\mathbf{x}} \in \mathbb{R}^d, \quad \varphi_{\text{geo}}^{[\mathbf{v}]^d}(\hat{\mathbf{x}}) = \mathbf{v}_j + \sum_{i=0, i \neq j}^{d} \hat{\mathcal{L}}_i^{1,d}(\hat{\mathbf{x}})(\mathbf{v}_i - \mathbf{v}_j), $$

$$ (9.38) \quad \forall \hat{\mathbf{x}} \in \mathbb{R}^d, \quad \varphi_{\text{geo}}^{[\mathbf{v}]^d}(\hat{\mathbf{x}}) = \mathbf{v}_0 + \sum_{i=1}^{d} \hat{x}_i(\mathbf{v}_i - \mathbf{v}_0) = \mathbf{v}_0 + A_{\text{geo}}^{[\mathbf{v}]^d} \hat{\mathbf{x}}, $$

$$ (9.39) \quad \forall i \in [0..d], \quad \varphi_{\text{geo}}^{[\mathbf{v}]^d}(\hat{\mathbf{v}}_i) = \mathbf{v}_i, $$

$$ (9.40) \qquad \varphi_{\text{geo}}^{[\mathbf{v}]^d}(\hat{\mathbf{K}}^d) = \mathbf{K}^{[\mathbf{v}]^d}. $$

Moreover, if $[\mathbf{v}]^d$ is affinely independent, then $\varphi_{\text{geo}}^{[\mathbf{v}]^d}$ is bijective, $(\varphi_{\text{geo}}^{[\mathbf{v}]^d})^{-1}$ is affine, and

$$ (9.41) \quad \forall \mathbf{x} \in \mathbb{R}^d, \quad (\varphi_{\text{geo}}^{[\mathbf{v}]^d})^{-1}(\mathbf{x}) = (A_{\text{geo}}^{[\mathbf{v}]^d})^{-1}(\mathbf{x} - \mathbf{v}_0), $$

$$ (9.42) \quad \forall i \in [0..d], \quad (\varphi_{\text{geo}}^{[\mathbf{v}]^d})^{-1}(\mathbf{v}_i) = \hat{\mathbf{v}}_i. $$

*Proof. Identities (9.37) and (9.38), affine.* Let $j \in [0..d]$. Then, from Definition 1547 (geometric mapping), Lemma 1543 (reference Lagrange polynomials is basis of $\mathbb{P}_1^d$, (9.33), extracting $j$), and ring properties of $\mathbb{R}$, identity (9.37) holds. Thus, from (9.37) (with $j \stackrel{\text{def.}}{=} 0$), Definition 1541 (reference Lagrange polynomials of $\mathbb{P}_1^d$, (9.30)), the columnwise rule for matrix-vector product, commutative ring properties of $\mathbb{R}$, and Definition 1388 (affine map, with $\phi \stackrel{\text{def.}}{=} A_{\text{geo}}^{[\mathbf{v}]^d}$ and $\mathbf{y}_0 \stackrel{\text{def.}}{=} \mathbf{v}_0$), identities (9.38) hold, and $\varphi_{\text{geo}}^{[\mathbf{v}]^d}$ is affine.

**Identity (9.39).** Direct consequence of Definition 1547 (geometric mapping), and Lemma 1543 (reference Lagrange polynomials is basis of $\mathbb{P}_1^d$, (9.32)).

**Identity (9.40).**

$\varphi_{\text{geo}}^{[\mathbf{v}]^d}(\hat{\mathbf{K}}^d) \subset \mathbf{K}^{[\mathbf{v}]^d}$. Let $\hat{\mathbf{x}} \in \hat{\mathbf{K}}^d$. Then, from Definition 1541 (reference Lagrange polynomials of $\mathbb{P}_1^d$), Definition 1437 (reference simplex), ordered field properties of $\mathbb{R}$, Lemma 1543 (reference Lagrange polynomials is basis of $\mathbb{P}_1^d$, (9.33)), Definition 1547 (geometric mapping), and Definition 1440 (simplex), we have, for all $i \in [1..d]$, $\hat{\mathcal{L}}_i^{1,d}(\hat{\mathbf{x}}) = \hat{x}_i \ge 0$, and $\hat{\mathcal{L}}_0^{1,d}(\hat{\mathbf{x}}) = 1 - \sum_{i=0}^d \hat{\mathcal{L}}_i^{1,d} \ge 0$,
thus $\varphi_{\text{geo}}^{[\mathbf{v}]^d}(\hat{\mathbf{x}})$ is a convex combination of the $\mathbf{v}_i$'s, and it belongs to $K^{[\mathbf{v}]^d}$.

$K^{[\mathbf{v}]^d} \subset \varphi_{\text{geo}}^{[\mathbf{v}]^d}(\hat{\mathbf{K}}^d)$. Let $\mathbf{x} \in K^{[\mathbf{v}]^d}$. Then, from Definition 1440 (simplex), there exists $(\mu_i)_{i \in [0..d]} \in \mathbb{R}$ such that, for all $i \in [0..d]$, $\mu_i \ge 0$, $\sum_{i=0}^d \mu_i = 1$ and $\mathbf{x} = \sum_{i=0}^d \mu_i \mathbf{v}_i$.

Let $\hat{\mathbf{x}} \stackrel{\text{def.}}{=} (\mu_i)_{i \in [1..d]} \in \mathbb{R}^d$. Then, from Definition 1541 (reference Lagrange polynomials of $\mathbb{P}_1^d$), ordered field properties of $\mathbb{R}$, Definition 1437 (reference simplex), and Definition 1547 (geometric mapping), we have, for all $i \in [1..d]$, $\hat{\mathcal{L}}_i^{1,d}(\hat{\mathbf{x}}) = \mu_i \ge 0$ and $\hat{\mathcal{L}}_0^{1,d}(\hat{\mathbf{x}}) = 1 - \sum_{j=1}^d \mu_j = \mu_0 \ge 0$, thus $\hat{\mathbf{x}} \in \hat{\mathcal{K}}^d$ and $\varphi_{\text{geo}}^{[\mathbf{v}]^d}(\hat{\mathbf{x}}) = \mathbf{x}$, i.e. $\mathbf{x} \in \varphi_{\text{geo}}^{[\mathbf{v}]^d}(\hat{\mathcal{K}}^d)$.

Therefore, we have equality.

**Bijection, ($\varphi_{\text{geo}}^{[\mathbf{v}]^d}$)${}^{-1}$ affine, and identity (9.41).** Direct consequence of (9.38), Lemma 1406 (injective affine map is zero linear kernel), Definition 101 (kernel), Definition 1411 (affinely independent family), Lemma 1374 (injectivity or surjectivity and dimension implies bijectivity), and Lemma 1410 (inverse of affine map is affine map).

**Identity (9.42).** Direct consequence of (9.39). □

<figure><img src="image_23.png" /></figure>

**Lemma 1551 (differential of geometric mapping).**

Let $d \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ points in $\mathbb{R}^d$. Then, $\varphi_{\text{geo}}^{[\mathbf{v}]^d}$ belongs to $C^\infty(\mathbb{R}^d, \mathbb{R}^d)$, and we have

$$ (9.43) \qquad \forall \hat{\mathbf{x}} \in \mathbb{R}^d, \quad D\varphi_{\text{geo}}^{[\mathbf{v}]^d}(\hat{\mathbf{x}}) = A_{\text{geo}}^{[\mathbf{v}]^d}. $$

Moreover, if $[\mathbf{v}]^d$ is affinely independent, $(\varphi_{\text{geo}}^{[\mathbf{v}]^d})^{-1}$ also belongs to $C^\infty(\mathbb{R}^d, \mathbb{R}^d)$, and we have

$$ (9.44) \qquad \forall \mathbf{x} \in \mathbb{R}^d, \quad D(\varphi_{\text{geo}}^{[\mathbf{v}]^d})^{-1}(\mathbf{x}) = (A_{\text{geo}}^{[\mathbf{v}]^d})^{-1}. $$

**Proof.** Direct consequence of Lemma 1550 (properties of geometric mapping), affine maps in $\mathbb{R}^d$ are $C^\infty$, and the rules of derivation (thus affine maps have constant differential and the differential is the linear part). □

<figure><img src="image_24.png" /></figure>

### 9.3.3 Current affine Lagrange polynomials

**Lemma 1552 (Lagrange polynomials of $\mathbb{P}_1^d$).**

Let $d \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$. Let $i \in [0..d]$.

Then, $L_i^{[\mathbf{v}]^d} \stackrel{\text{def.}}{=} \hat{\mathcal{L}}_i^{1,d} \circ (\varphi_{\text{geo}}^{[\mathbf{v}]^d})^{-1}$ is well-defined, and we have $\hat{\mathcal{L}}_i^{1,d} = L_i^{[\mathbf{v}]^d} \circ \varphi_{\text{geo}}^{[\mathbf{v}]^d}$.

$L_i^{[\mathbf{v}]^d}$ is called the $i$-th Lagrange polynomial of $\mathbb{P}_1^d$ associated with points $[\mathbf{v}]^d$.

**Proof. Existence.** Direct consequence of Lemma 1550 (properties of geometric mapping).

**Property.** Direct consequence of associativity of composition of functions, and the definition of the inverse. □

<figure><img src="image_25.png" /></figure>

**Lemma 1553 (Lagrange polynomials of reference vertices are reference Lagrange polynomials of $\mathbb{P}_1^d$).**
Let $d \ge 1$. Let $i \in [0..d]$. Then, we have $L_i^{[\mathbf{v}]^d} = \hat{\mathcal{L}}_i^{1,d}$.

**Proof.** Direct consequence of Lemma 1552 (Lagrange polynomials of $\mathbb{P}_1^d$), and Lemma 1549 (reference geometric mapping is identity). □

<figure><img src="image_26.png" /></figure>
**Lemma 1554 (Lagrange polynomials is basis of $\mathbb{P}_1^d$).**

Let $d \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$.

Then, the Lagrange polynomials $(\mathcal{L}_i^{[\mathbf{v}]^d})_{i \in [0..d]}$ form a basis of $\mathbb{P}_1^d$, that satisfies

$$ (9.45) \qquad \forall i, j \in [0..d], \quad \mathcal{L}_i^{[\mathbf{v}]^d}(\mathbf{v}_j) = \delta_{ij}, $$

$$ (9.46) \qquad \sum_{i=0}^{d} \mathcal{L}_i^{[\mathbf{v}]^d} = 1. $$

*Proof.* In $\mathbb{P}_1^d$. Direct consequence of Lemma 1552 (Lagrange polynomials of $\mathbb{P}_1^d$), Lemma 1509 (constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$, thus $\mathbb{P}_1^d = \text{Aff}(\mathbb{R}^d, \mathbb{R})$), Lemma 1543 (reference Lagrange polynomials is basis of $\mathbb{P}_1^d$, thus $\hat{\mathbb{L}}_{i}^{1,d}$ is affine), Lemma 1550 (properties of geometric mapping, thus $(\varphi_{\text{geo}}^{[\mathbf{v}]^d})^{-1}$ is affine), and Lemma 1403 (affine maps are closed by composition).

**Identities (9.45) and (9.46).**

Direct consequence of Lemma 1552 (Lagrange polynomials of $\mathbb{P}_1^d$), Lemma 1550 (properties of geometric mapping, (9.42)), Lemma 1543 (reference Lagrange polynomials is basis of $\mathbb{P}_1^d$, (9.32) and (9.33)), and **distributivity of composition over addition.**

**Basis.** Let $(\lambda_i)_{i \in [0..d]} \in \mathbb{R}$, such that for all $\mathbf{x} \in \mathbb{R}^d$, $\sum_{i=0}^d \lambda_i \mathcal{L}_i^{[\mathbf{v}]^d}(\mathbf{x}) = 0$. Then, from (9.45), and **ring properties of $\mathbb{R}$**, taking $\mathbf{x} \stackrel{\text{def.}}{=} \mathbf{v}_j$ provides $\sum_{i=0}^d \lambda_i \delta_{ij} = \lambda_j = 0$. Thus, from **the definition of freedom**, Lemma 1525 (dimension of $\mathbb{P}_k^d$, with $k \stackrel{\text{def.}}{=} 1$), and Lemma 1376 (free family of dim elements is basis), the Lagrange polynomials form a basis of $\mathbb{P}_1^d$. $\square$

**Lemma 1555 (decomposition of $\mathbb{P}_1^d$ polynomial in Lagrange basis).**

Let $d \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$. Let $p \in \mathbb{P}_1^d$. Then, we have

$$ (9.47) \qquad p = \sum_{i=0}^{d} p(\mathbf{v}_i) \mathcal{L}_i^{[\mathbf{v}]^d}. $$

*Proof.* Direct consequence of Lemma 1554 (Lagrange polynomials is basis of $\mathbb{P}_1^d$, basis, then (9.45)), and **ring properties of $\mathbb{R}$**. $\square$

**Lemma 1556 (differential of Lagrange polynomials $\mathbb{P}_1^d$).**

Let $d \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$. Let $\mathbf{x} \in \mathbb{R}^d$. Let $i \in [0..d]$.

Then, we have $\mathcal{L}_i^{[\mathbf{v}]^d} \in \mathcal{C}^\infty(\mathbb{R}^d, \mathbb{R})$ and

$$ (9.48) \qquad (i=0) \qquad \mathrm{D}\mathcal{L}_0^{[\mathbf{v}]^d}(\mathbf{x}) = -\sum_{j=1}^{d} \frac{(A_{\text{geo}}^{[\mathbf{v}]^d})^{-1}}{j} \in \mathcal{M}_{1,d}(\mathbb{R}), $$

$$ (9.49) \qquad (i \in [1..d]) \qquad \mathrm{D}\mathcal{L}_i^{[\mathbf{v}]^d}(\mathbf{x}) = (A_{\text{geo}}^{[\mathbf{v}]^d})^{-1} \in \mathcal{M}_{1,d}(\mathbb{R}), $$

where, we recall that for any matrix $B$, $\underline{B}_i$ denotes its $i$-th line.

*Proof.* Direct consequence of Lemma 1554 (Lagrange polynomials is basis of $\mathbb{P}_1^d$), Lemma 1509 (constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$, thus Lagrange polynomials are affine maps), **affine maps in $\mathbb{R}^d$ are $\mathcal{C}^\infty$, Lemma 1545 (differential of reference Lagrange polynomials)**, Lemma 1551 (differential of geometric mapping), and **the rules derivation for composition**. $\square$
### 9.3.4 Nontrivial current simplex

**Lemma 1557 (nontrivial simplex).** Let $d \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$. Then, $K[\mathbf{v}]^d$ has a nonempty interior. It is then said to be nontrivial.

*Proof.* From Lemma 1439 (nontrivial reference simplex), and the definition of the interior, let $\hat{U}$ be a nonempty open set included in $\hat{K}^d$. Then, from Lemma 1551 (differential of geometric mapping), $C^\infty$ is continuous (thus $\varphi_{\text{geo}}[\mathbf{v}]^d$ is an homeomorphism), and homeomorphisms are open maps, $U \stackrel{\text{def.}}{=} \varphi_{\text{geo}}[\mathbf{v}]^d(\hat{U})$ is open. Therefore, from image of $\emptyset$ is $\emptyset$ (contrapositive), image is nondecreasing, Lemma 1550 (properties of geometric mapping, (9.40)), and the definition of the interior, we have $\emptyset \ne U \subset K[\mathbf{v}]^d$, and $K[\mathbf{v}]^d$ has a nonempty interior. $\square$

## 9.4 Barycentric coordinates

*Remark* 1558. In the next lemma, the uniqueness of the barycentric coordinates actually makes them functions of the point $\mathbf{x}$. In particular, in (9.51), they do not depend on the choice for $j$.

**Lemma 1559 (barycentric coordinate).**

Let $d \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$. Let $\mathbf{x} \in \mathbb{R}^d$.

Then, there exists unique $(\lambda_i^{[\mathbf{v}]^d})_{i \in [0..d]} \in \mathbb{R}$ satisfying the next two equivalent decompositions

$$ (9.50) \qquad \mathbf{x} = \sum_{i=0}^{d} \lambda_i^{[\mathbf{v}]^d} \mathbf{v}_i \qquad \text{with } \sum_{i=0}^{d} \lambda_i^{[\mathbf{v}]^d} = 1, $$

$$ (9.51) \qquad \forall j \in [0..d], \quad \mathbf{x} = \mathbf{v}_j + \sum_{i=0, i \neq j}^{d} \lambda_i^{[\mathbf{v}]^d} (\mathbf{v}_i - \mathbf{v}_j) \qquad \text{with } \lambda_j^{[\mathbf{v}]^d} = 1 - \sum_{i=0, i \neq j}^{d} \lambda_i^{[\mathbf{v}]^d}. $$

The $\lambda_i^{[\mathbf{v}]^d}$'s are called barycentric coordinates of $\mathbf{x}$ with respect to $[\mathbf{v}]^d$, they are functions of $\mathbf{x}$.

*Proof.* **Existence.** Let $j \in [0..d]$. Then, from Lemma 81 (closed under vector operations is subspace, thus $\mathbf{x} - \mathbf{v}_j$ belongs to the vector space $\mathbb{R}^d$), Lemma 1412 (equivalent definition of affinely independent family), Lemma 1376 (free family of dim elements is basis), and the definition of basis, there exist unique real numbers $(\mu_i^j)_{i \in [0..d] \setminus \{j\}}$ such that $\mathbf{x} - \mathbf{v}_j = \sum_{i=0, i \neq j} \mu_i^j (\mathbf{v}_i - \mathbf{v}_j)$, i.e. the decomposition (9.51) exists, and is unique (but still depends on $j$).

Let $\mu_j^j \stackrel{\text{def.}}{=} 1 - \sum_{i=0, i \neq j} \mu_i^j$. Then, from commutative ring properties of $\mathbb{R}$, we have $\mathbf{x} = \sum_{i=0} \mu_i^j \mathbf{v}_i$ and $\sum_{i=0}^d \mu_i^j = 1$, i.e. the decomposition (9.50) exists.

**Uniqueness.** Let $(\lambda_i)_{i \in [0..d]}$, $(\mu_i)_{i \in [0..d]} \in \mathbb{R}$, such that

$$ \mathbf{x} = \sum_{i=0}^{d} \lambda_i \mathbf{v}_i = \sum_{i=0}^{d} \mu_i \mathbf{v}_i \quad \text{and} \quad \sum_{i=0}^{d} \lambda_i = \sum_{i=0}^{d} \mu_i = 1. $$

For all $i \in [0..d]$, let $\xi_i \stackrel{\text{def.}}{=} \lambda_i - \mu_i$. Then, from ring properties of $\mathbb{R}$, we have successively $\sum_{i=0}^d \xi_i \mathbf{v}_i = 0$, $\sum_{i=0}^d \xi_i = 0$, $\xi_0 = -\sum_{i=1}^d \xi_i$, and $\sum_{i=1}^d \xi_i (\mathbf{v}_i - \mathbf{v}_0) = 0$. Thus, from Definition 1411 (affinely independent family), the definition of freedom, and additive group properties of $\mathbb{R}$, we have, for all $i \in [1..d]$, $\xi_i = 0$, and $\xi_0 = 0$. Therefore, the decomposition (9.50) is unique.

**Equivalence and independence on $j$**.

Let $j \in [0..d]$. Then, from (9.50), and ring properties of $\mathbb{R}$, we have

$$ \mathbf{x} = \left( 1 - \sum_{i=0, i \neq j} \lambda_i^{[\mathbf{v}]^d} \right) \mathbf{v}_j + \sum_{i=0, i \neq j} \lambda_i^{[\mathbf{v}]^d} \mathbf{v}_i, $$
and thus $\mathbf{x} - \mathbf{v}_j = \sum_{i=0, i \neq j}^{d} \lambda_i^{[\mathbf{v}]^d}(\mathbf{v}_i - \mathbf{v}_j)$, i.e. equivalence of the two decompositions. Moreover, from the uniqueness of the $\mu_i^j$'s defined above, we have, for all $i \in [0..d] \setminus \{j\}$, $\mu_i^j = \lambda_i^{[\mathbf{v}]^d}$, i.e. the decomposition (9.51) is independent on $j$. $\square$

**Lemma 1560 (Lagrange polynomials of $\mathbb{P}_1^d$ are barycentric coordinate).**

Let $d \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$.

Then, barycentric coordinates and Lagrange polynomials coincide, for all $i \in [0..d]$, $\lambda_i^{[\mathbf{v}]^d} = \mathcal{L}_i^{[\mathbf{v}]^d}$.

Thus, we have, for all $\mathbf{x} \in \mathbb{R}^d$, $\mathbf{x} = \sum_{i=0}^{d} \mathcal{L}_i^{[\mathbf{v}]^d}(\mathbf{x}) \mathbf{v}_i$ with $\sum_{i=0}^{d} \mathcal{L}_i^{[\mathbf{v}]^d}(\mathbf{x}) = 1$.

Moreover, let $\lambda^{[\mathbf{v}]^d} \stackrel{\text{def.}}{=} (\lambda_i^{[\mathbf{v}]^d})_{i \in [1..d]}$. Then, we have $\lambda^{[\mathbf{v}]^d} = (\varphi_{\text{geo}}^{[\mathbf{v}]^d})^{-1} \in \text{Aff}(\mathbb{R}^d, \mathbb{R}^d)$, i.e.

$$ (9.52) \quad \forall \mathbf{x} \in \mathbb{R}^d, \quad \mathbf{x} = \varphi_{\text{geo}}^{[\mathbf{v}]^d}(\lambda^{[\mathbf{v}]^d}(\mathbf{x})), \quad \text{or equivalently} \quad \forall \hat{\mathbf{x}} \in \mathbb{R}^d, \quad \hat{\mathbf{x}} = \lambda^{[\mathbf{v}]^d}(\varphi_{\text{geo}}^{[\mathbf{v}]^d}(\hat{\mathbf{x}})). $$

**Proof. (i) Equality $\lambda^{[\mathbf{v}]^d} = (\varphi_{\text{geo}}^{[\mathbf{v}]^d})^{-1}$.** Let $\mathbf{x} \in \mathbb{R}^d$. Then, from Lemma 1559 (barycentric coordinate, (9.51) with $j \stackrel{\text{def.}}{=} 0$), additive group properties of $\mathbb{R}$, the columnwise rule for matrix-vector product, the rules of matrix-vector product (inverse matrix), and Lemma 1550 (properties of geometric mapping), we have

$$ \mathbf{x} - \mathbf{v}_0 = \sum_{i=1}^{d} \lambda_i^{[\mathbf{v}]^d}(\mathbf{x})(\mathbf{v}_i - \mathbf{v}_0) = A_{\text{geo}}^{[\mathbf{v}]^d} \lambda^{[\mathbf{v}]^d}(\mathbf{x}), $$

$$ \lambda^{[\mathbf{v}]^d}(\mathbf{x}) = (A_{\text{geo}}^{[\mathbf{v}]^d})^{-1}(\mathbf{x} - \mathbf{v}_0) = (\varphi_{\text{geo}}^{[\mathbf{v}]^d})^{-1}(\mathbf{x}), $$

and thus $\lambda^{[\mathbf{v}]^d} = (\varphi_{\text{geo}}^{[\mathbf{v}]^d})^{-1} \in \text{Aff}(\mathbb{R}^d, \mathbb{R}^d)$.

(ii) Identities (9.52). Direct consequence of (i), and the definition of the inverse.

(iii) Identity $\lambda_i^{[\mathbf{v}]^d} = \mathcal{L}_i^{[\mathbf{v}]^d}$.

Let $\mathbf{x} \in \mathbb{R}^d$. Then, from Lemma 1559 (barycentric coordinate, (9.50), then uniqueness), (ii), Definition 1547 (geometric mapping), (i), and Lemma 1552 (Lagrange polynomials of $\mathbb{P}_1^d$), we have

$$ \sum_{i=0}^{d} \lambda_i^{[\mathbf{v}]^d}(\mathbf{x}) \mathbf{v}_i = \mathbf{x} = \varphi_{\text{geo}}^{[\mathbf{v}]^d}(\lambda^{[\mathbf{v}]^d}(\mathbf{x})) $$

$$ = \sum_{i=0}^{d} \hat{\mathcal{L}}_{i}^{1,d}(\lambda^{[\mathbf{v}]^d}(\mathbf{x})) \mathbf{v}_i = \sum_{i=0}^{d} \hat{\mathcal{L}}_{i}^{1,d}\left(\left(\varphi_{\text{geo}}^{[\mathbf{v}]^d}\right)^{-1}(\mathbf{x})\right) \mathbf{v}_i = \sum_{i=0}^{d} \mathcal{L}_i^{[\mathbf{v}]^d}(\mathbf{x}) \mathbf{v}_i, $$

and thus, for all $i \in [0..d]$, $\lambda_i^{[\mathbf{v}]^d} = \mathcal{L}_i^{[\mathbf{v}]^d}$. $\square$

**Lemma 1561 (decomposition of $\mathbb{P}_1^d$ polynomial with barycentric coordinates).**

Let $d \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$. Let $p \in \mathbb{P}_1^d$. Then, we have

$$ (9.53) \qquad p = \sum_{i=0}^{d} p(\mathbf{v}_i) \lambda_i^{[\mathbf{v}]^d}. $$

**Proof.** Direct consequence of Lemma 1555 (decomposition of $\mathbb{P}_1^d$ polynomial in Lagrange basis), and Lemma 1560 (Lagrange polynomials of $\mathbb{P}_1^d$ are barycentric coordinate). $\square$
## 9.5 Hyperplanes and $l$-faces

### 9.5.1 Hyperplanes and $d-1$-faces

**Definition 1562 (face hyperplane).**

Let $d \ge 1$. Let $[[v]]^d$ be $d+1$ points in $\mathbb{R}^d$. Let $i \in [0..d]$. The $i$-th face hyperplane opposite vertex $v_i$ is denoted $\mathcal{H}_i^{[[v]]^d}$, and is defined as the affine subspace of $\mathbb{R}^d$

$$ (9.54) \qquad (i = 0) \qquad \mathcal{H}_0^{[[v]]^d} \stackrel{\text{def.}}{=} \mathbf{v}_1 + \text{span} (\mathbf{v}_j - \mathbf{v}_1)_{j \in [2..d]}, $$

$$ (9.55) \qquad (i \in [1..d]) \qquad \mathcal{H}_i^{[[v]]^d} \stackrel{\text{def.}}{=} \mathbf{v}_0 + \text{span} (\mathbf{v}_j - \mathbf{v}_0)_{j \in [1..d] \setminus \{i\}}. $$

For the reference vertices $[[\hat{v}]]^d$, the $i$-th reference face hyperplane is denoted $\hat{\mathcal{H}}_i^{[\hat{v}]} \stackrel{\text{def.}}{=} \mathcal{H}_i^{[[\hat{v}]]^d}$.

<figure><img src="image_27.png" /><figcaption>Figure 9.5: Geometric transformation $\varphi_{\text{geo}}^{[[v]]^d}$ in the case $d = k = 3$.</figcaption></figure>

The reference simplex $\hat{\mathcal{R}}^d$ is mapped onto the current simplex $K^{[[v]]^d}$, and reference nodes $\hat{a}_\alpha$ onto current nodes $a_\alpha$, see Lemmas 1550 and 1604. For instance, we show that the reference hyperplane $\mathcal{H}_0^d$ containing the face $\hat{\mathcal{H}}_0^d$ is mapped onto $\mathcal{H}_0^{[[v]]^d}$ containing the face $\mathcal{H}_0^{[[v]]^d}$. The nodes in these two faces are colored in order to help see the correspondence: for all $\alpha \in C_3^3$, we have $a_\alpha = \varphi_{\text{geo}}^{[[v]]^d}(\hat{a}_\alpha)$.

**Lemma 1563 (equivalent definition of face hyperplane).** Let $d \ge 1$. Let $[[v]]^d$ be $d+1$ points in $\mathbb{R}^d$. Let $i \in [0..d]$. Then, the $i$-th face hyperplane is characterized by

$$ (9.56) \qquad \mathcal{H}_i^{[[v]]^d} = \left\{ \sum_{j=0, j \neq i}^{d} \mu_j v_j \in \mathbb{R}^d \mid \forall j \in [0..d] \setminus \{i\}, \ \mu_j \in \mathbb{R} \ \wedge \ \sum_{j=0, j \neq i}^{d} \mu_j = 1 \right\}. $$

Moreover, if $[[v]]^d$ is affinely independent, then we have

$$ (9.57) \qquad \mathcal{H}_i^{[[v]]^d} = \ker (\lambda_i^{[[v]]^d}), $$

where $\lambda_i^{[[v]]^d}$ is the $i$-th barycentric coordinate application.
**Proof. Identity (9.56).** Direct consequence of Lemma 1387 (barycenter closure is affine subspace, with $n \stackrel{\text{def.}}{=} d - 1 \ge 0$, and $\mathbf{v}_j$ for $j \in [0..d] \setminus \{i\}$).

**Identity (9.57).** Case $i = 0$. Then, from Definition 1562 (face hyperplane), ring properties of $\mathbb{R}$, Lemma 1559 (barycentric coordinate, (9.51) with $j \stackrel{\text{def.}}{=} 1$), we have

$$ 
\begin{aligned}
\mathbf{x} \in \mathcal{H}_0^{[\mathbf{v}]^d} &\iff \exists (\tilde{\mu}_j)_{j \in [2..d]} \in \mathbb{R}, \quad \mathbf{x} = \mathbf{v}_1 + 0(\mathbf{v}_0 - \mathbf{v}_1) + \sum_{j=2}^d \tilde{\mu}_j (\mathbf{v}_j - \mathbf{v}_1) \\
&\iff \lambda_0^{[\mathbf{v}]^d}(\mathbf{x}) = 0.
\end{aligned}
$$

**Case $i \in [1..d]$.** Then, from Definition 1562 (face hyperplane), ring properties of $\mathbb{R}$, Lemma 1559 (barycentric coordinate, (9.51) with $j \stackrel{\text{def.}}{=} 0$), we have

$$ 
\begin{aligned}
\mathbf{x} \in \mathcal{H}_i^{[\mathbf{v}]^d} &\iff \exists (\tilde{\mu}_j)_{j \in [1..d] \setminus \{i\}} \in \mathbb{R}, \quad \mathbf{x} = \mathbf{v}_0 + 0(\mathbf{v}_i - \mathbf{v}_0) + \sum_{j=1, j \ne i}^d \tilde{\mu}_j (\mathbf{v}_j - \mathbf{v}_0) \\
&\iff \lambda_i^{[\mathbf{v}]^d}(\mathbf{x}) = 0.
\end{aligned}
$$

Therefore, from Definition 101 (kernel), the identity always holds. $\square$

**Lemma 1564 (reference face hyperplane).**

Let $d \ge 1$. Let $i \in [0..d]$. Then, we have

$$ (9.58) \qquad (i = 0) \qquad \hat{\mathcal{H}}_0^d = \left\{ \hat{\mathbf{x}} \in \mathbb{R}^d \mid \sum_{j=1}^d \hat{x}_j = 1 \right\} = \ker (\hat{\mathcal{L}}_0^{1,d}), $$

$$ (9.59) \qquad (i \in [1..d]) \qquad \hat{\mathcal{H}}_i^d = \{ \hat{\mathbf{x}} \in \mathbb{R}^d \mid \hat{x}_i = 0 \} = \ker (\hat{\mathcal{L}}_i^{1,d}). $$

**Proof.** Direct consequence of Lemma 1436 (family of reference points is affinely independent), Lemma 1563 (equivalent definition of face hyperplane, (9.57)), Lemma 1560 (Lagrange polynomials of $\mathbb{P}_1^d$ are barycentric coordinate), Lemma 1553 (Lagrange polynomials of reference vertices are reference Lagrange polynomials of $\mathbb{P}_1^d$), Definition 1541 (reference Lagrange polynomials of $\mathbb{P}_1^d$), and Definition 101 (kernel). $\square$

**Lemma 1565 (face hyperplane is image of reference face hyperplane).**

Let $d \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$. Let $i \in [0..d]$. Then, we have

$$ (9.60) \qquad \mathcal{H}_i^{[\mathbf{v}]^d} = \varphi_{\text{geo}}^{[\mathbf{v}]^d} (\hat{\mathcal{H}}_i^d). $$

**Proof.** Case $i = 0$. Then, from Definition 1562 (face hyperplane), ring properties of $\mathbb{R}$, Definition 1541 (reference Lagrange polynomials of $\mathbb{P}_1^d$, thus $\hat{\mathcal{L}}_j^{1,d}(\hat{\mu}) = \mu_j$ for all $j \in [2..d]$, $\hat{\mathcal{L}}_1^{1,d}(\hat{\mu})$ equals $(1 - \sum_{j=2}^d \mu_j)$ and $\hat{\mathcal{L}}_0^{1,d}(\hat{\mu}) = 0$), Definition 1547 (geometric mapping), and Definition 1434
(family of reference points), we have

$$
\begin{align*}
\mathbf{x} \in \mathcal{H}_0^{[\mathbf{v}]^d} &\iff \exists(\mu_j)_{j \in [2..d]} \in \mathbb{R}, \quad \mathbf{x} = \mathbf{v}_1 + \sum_{j=2}^d \mu_j (\mathbf{v}_j - \mathbf{v}_1) \\
&\iff \exists(\mu_j)_{j \in [2..d]} \in \mathbb{R}, \quad \mathbf{x} = 0\mathbf{v}_0 + \left(1 - \sum_{j=2}^d \mu_j\right) \mathbf{v}_1 + \sum_{j=2}^d \mu_j \mathbf{v}_j \\
&\iff \exists(\mu_j)_{j \in [2..d]} \in \mathbb{R}, \quad \mu_1 \stackrel{\text{def.}}{=} 1 - \sum_{j=2}^d \mu_j \wedge \mathbf{x} = \left(1 - \sum_{j=1}^d \mu_j\right) \mathbf{v}_0 + \sum_{j=1}^d \mu_j \mathbf{v}_j \\
&\iff \exists(\mu_j)_{j \in [2..d]} \in \mathbb{R}, \quad \hat{\mu} \stackrel{\text{def.}}{=} \left(1 - \sum_{j=2}^d \mu_j\right) \mathbf{e}_1 + \sum_{j=2}^d \mu_j \mathbf{e}_j \in \mathbb{R}^d \wedge \mathbf{x} = \varphi_{\text{geo}}^{[\mathbf{v}]^d}(\hat{\mu}) \\
&\iff \exists(\mu_j)_{j \in [2..d]} \in \mathbb{R}, \quad \hat{\mu} \stackrel{\text{def.}}{=} \hat{\mathbf{v}}_1 + \sum_{j=2}^d \mu_j (\hat{\mathbf{v}}_j - \hat{\mathbf{v}}_1) \in \hat{\mathcal{H}}_0^d \wedge \mathbf{x} = \varphi_{\text{geo}}^{[\mathbf{v}]^d}(\hat{\mu}).
\end{align*}
$$

Therefore, we have the equality.

Case $i \in [1..d]$. Then, from Definition 1562 (face hyperplane), ring properties of $\mathbb{R}$, Definition 1541 (reference Lagrange polynomials of $\mathbb{P}_1^d$, thus $\hat{\mathcal{L}}_j^{1,d}(\hat{\mu}) = \mu_j$ for all $j \in [1..d] \setminus \{i\}$, $\hat{\mathcal{L}}_0^{1,d}(\hat{\mu}) = (1 - \sum_{l=1, l \neq i}^d \mu_l)$ and $\hat{\mathcal{L}}_i^{1,d}(\hat{\mu}) = 0$), Definition 1547 (geometric mapping), and Definition 1434 (family of reference points), we have

$$
\begin{align*}
\mathbf{x} \in \mathcal{H}_i^{[\mathbf{v}]^d} &\iff \exists (\mu_j)_{j \in [1..d] \setminus \{i\}} \in \mathbb{R}, \quad \mathbf{x} = \mathbf{v}_0 + \sum_{j=1, j \neq i}^{d} \mu_j (\mathbf{v}_j - \mathbf{v}_0) \\
&\iff \exists (\mu_j)_{j \in [1..d] \setminus \{i\}} \in \mathbb{R}, \quad \mathbf{x} = \left(1 - \sum_{j=1, j \neq i}^{d} \mu_j\right) \mathbf{v}_0 + 0\mathbf{v}_i + \sum_{j=1, j \neq i}^{d} \mu_j \mathbf{v}_j \\
&\iff \exists (\mu_j)_{j \in [1..d] \setminus \{i\}} \in \mathbb{R}, \quad \mu_i \stackrel{\text{def.}}{=} 0 \wedge \mathbf{x} = \left(1 - \sum_{j=1}^{d} \mu_j\right) \mathbf{v}_0 + \sum_{j=1}^{d} \mu_j \mathbf{v}_j \\
&\iff \exists (\mu_j)_{j \in [1..d] \setminus \{i\}} \in \mathbb{R}, \quad \hat{\mu} \stackrel{\text{def.}}{=} 0\mathbf{e}_i + \sum_{j=1, j \neq i}^{d} \mu_j \mathbf{e}_j \in \mathbb{R}^d \wedge \mathbf{x} = \varphi_{\text{geo}}^{[\mathbf{v}]^d}(\hat{\mu}) \\
&\iff \exists (\mu_j)_{j \in [1..d] \setminus \{i\}} \in \mathbb{R}, \quad \hat{\mu} \stackrel{\text{def.}}{=} \hat{\mathbf{v}}_0 + \sum_{j=1, j \neq i}^{d} \mu_j (\hat{\mathbf{v}}_j - \hat{\mathbf{v}}_0) \in \hat{\mathcal{H}}_i^d \wedge \mathbf{x} = \varphi_{\text{geo}}^{[\mathbf{v}]^d}(\hat{\mu}).
\end{align*}
$$

Therefore, we have the equality.

<figure><img src="image_28.png" /></figure>

**Definition 1566 (hyperface).**

Let $d \ge 1$. Let $[[\mathbf{v}]^d$ be $d+1$ points in $\mathbb{R}^d$. Let $i \in [0..d]$. The $i$-th hyperface opposite vertex $\mathbf{v}_i$, having the vertices $(\mathbf{v}_j)_{j \in [0..d] \setminus \{i\}}$, is denoted $H_i^{[\mathbf{v}]^d}$, and is defined by

$$
(9.61) \quad H_i^{[\mathbf{v}]^d} \stackrel{\text{def.}}{=} \left\{ \sum_{j=0, j \neq i}^{d} \mu_j \mathbf{v}_j \in \mathbb{R}^d \;\middle|\; \forall j \in [0..d] \setminus \{i\}, \quad \mu_j \ge 0 \quad \wedge \quad \sum_{j=0, j \neq i}^{d} \mu_j = 1 \right\}.
$$

**Lemma 1567 (hyperface is included in face hyperplane).**

Let $d \ge 1$. Let $[[\mathbf{v}]^d$ be $d+1$ points in $\mathbb{R}^d$. Let $i \in [0..d]$. Then, we have $H_i^{[\mathbf{v}]^d} \subset H_i^{[\mathbf{v}]^d}$.

*Proof.* Direct consequence of Definition 1566 (hyperface), and Lemma 1563 (equivalent definition of face hyperplane). $\square$

<figure><img src="image_29.png" /></figure>
### 9.5.2 l-face affine spaces and l-faces

**Definition 1568 (l-face affine space).** Let $d \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ points in $\mathbb{R}^d$. Let $l \in [0..d]$. Let $\pi^l$ be an injective map from $[0..l]$ to $[0..d]$. The l-face affine space associated with $[\mathbf{v}]^d$ and having the vertices $[\mathbf{v}_{\pi^l(j)}]^{l,d} \stackrel{\text{def.}}{=} (\mathbf{v}_{\pi^l(j)})_{j \in [0..l]}$ is denoted $\mathcal{F}_{\pi^l}^{[\mathbf{v}]^d}$, and is defined by

$$ (9.62) \qquad \mathcal{F}_{\pi^l}^{[\mathbf{v}]^d} \stackrel{\text{def.}}{=} \left\{ \sum_{j=0}^{l} \mu_j \mathbf{v}_{\pi^l(j)} \in \mathbb{R}^d \;\middle|\; \forall j \in [0..l], \quad \mu_j \in \mathbb{R} \quad \wedge \quad \sum_{j=0}^{l} \mu_j = 1 \right\}. $$

**Lemma 1569 (equivalent definition of l-face affine space).** Let $d \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ points in $\mathbb{R}^d$. Let $l \in [0..d]$. Let $\pi^l$ be an injective map from $[0..l]$ to $[0..d]$. Then, $\mathcal{F}_{\pi^l}^{[\mathbf{v}]^d}$ is an affine space, that can be equivalently defined for any $i \in [0..l]$ by

$$ (9.63) \qquad \mathcal{F}_{\pi^l}^{[\mathbf{v}]^d} = \mathbf{v}_{\pi^l(i)} + \operatorname{span} (\mathbf{v}_{\pi^l(j)} - \mathbf{v}_{\pi^l(i)})_{j \in [0..l] \setminus \{i\}}. $$

*Proof.* Direct consequence of Definition 1568 (l-face affine space), and Lemma 1387 (barycenter closure is affine subspace, with $n \stackrel{\text{def.}}{=} l$ and $\mathbf{v}_j \stackrel{\text{def.}}{=} \mathbf{v}_{\pi^l(j)}$ for $j \in [0..l]$). $\square$

**Lemma 1570 (d-face affine space is full space).** Let $d \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$. Let $\pi^d$ be an injective map from $[0..d]$ to $[0..d]$. Then, $\mathcal{F}_{\pi^d}^{[\mathbf{v}]^d} = \mathbb{R}^d$.

*Proof.* Direct consequence of Lemma 1569 (equivalent definition of l-face affine space, with $i \stackrel{\text{def.}}{=} 0$), Lemma 1414 (affinely independent family is closed by sub-family, with $j \stackrel{\text{def.}}{=} 0$), Definition 1411 (affinely independent family, with $[\mathbf{v}_{\pi^d}]^d$), Lemma 1376 (free family of dim elements is basis), and $\mathbb{R}^d$ is affine space with any origin. $\square$

**Lemma 1571 (0-face affine space is vertex).** Let $d \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ points in $\mathbb{R}^d$. Let $\pi^0$ be a map from $\{0\}$ to $[0..d]$. Then, $\mathcal{F}_{\pi^0}^{[\mathbf{v}]^d}$ is the affine space $\{\mathbf{v}_{\pi^0(0)}\} = \mathbf{v}_{\pi^0(0)} + \{0\}$.

*Proof.* Direct consequence of the injectivity of a function defined on singleton, and Lemma 1569 (equivalent definition of l-face affine space, with $l \stackrel{\text{def.}}{=} 0$). $\square$

**Definition 1572 (l-face).** Let $d \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ points in $\mathbb{R}^d$. Let $l \in [0..d]$. Let $\pi^l$ an injective map from $[0..l]$ to $[0..d]$. The l-face associated with $[\mathbf{v}]^d$ and having the vertices $[\mathbf{v}_{\pi^l}]^{l,d}$ is denoted $F_{\pi^l}^{[\mathbf{v}]^d}$, and is defined by

$$ (9.64) \qquad F_{\pi^l}^{[\mathbf{v}]^d} \stackrel{\text{def.}}{=} \left\{ \sum_{j=0}^{l} \mu_j \mathbf{v}_{\pi^l(j)} \in \mathbb{R}^d \;\middle|\; \forall j \in [0..l], \quad \mu_j \ge 0 \quad \wedge \quad \sum_{j=0}^{l} \mu_j = 1 \right\}. $$

**Lemma 1573 (l-face is included in l-face affine space).** Let $d \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ points in $\mathbb{R}^d$. Let $l \in [0..d]$. Let $\pi^l$ an injective map from $[0..l]$ to $[0..d]$. Then, we have $F_{\pi^l}^{[\mathbf{v}]^d} \subset \mathcal{F}_{\pi^l}^{[\mathbf{v}]^d}$.

*Proof.* Direct consequence of Definition 1572 (l-face), and Definition 1568 (l-face affine space). $\square$

**Lemma 1574 (d-face is simplex).** Let $d \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ points in $\mathbb{R}^d$. Let $\pi^d$ be an injective map from $[0..d]$ to $[0..d]$. Then, $\pi^d$ is a bijective permutation of $[0..d]$, and we have $F_{\pi^d}^{[\mathbf{v}]^d} = K^{[\mathbf{v}]^d}$.
**Proof. Bijectivity.**

Direct consequence of **the fact that injectivity and cardinal equality imply bijectivity.**

**Equality.** From Definition 1572 (l-face), and Definition 1440 (simplex), we have

$$F_{\pi^d}[\mathbf{v}]^d = \left\{ \sum_{j=0}^{d} \mu_j \mathbf{v}_{\pi^d(j)} \in \mathbb{R}^d \;\middle|\; \forall j \in [0..d], \quad \mu_j \ge 0 \quad \wedge \quad \sum_{j=0}^{d} \mu_j = 1 \right\} \\ = \left\{ \sum_{i=0}^{d} \lambda_i \mathbf{v}_i \in \mathbb{R}^d \;\middle|\; \forall i \in [0..d], \quad \lambda_i \ge 0 \quad \wedge \quad \sum_{i=0}^{d} \lambda_i = 1 \right\} = K[\mathbf{v}]^d,$$

with $i \stackrel{\text{def.}}{=} \pi^d(j) \Leftrightarrow j = (\pi^d)^{-1}(i)$ ($\pi^d$ bijective in $[0..d]$) and $\lambda_i \stackrel{\text{def.}}{=} \mu_{(\pi^d)^{-1}(i)}$. $\square$

*Remark 1575.* In the next lemma, the injection $\theta_i^{d-1}$ from $[0..d-1]$ to $[0..d]$ is the “jump” enumeration function defined in Lemma 1368. The $(d-1)$-face is used to build the geometric hyperface mapping in Lemma 1584, see also Figure 9.6.

**Lemma 1576** ((d − 1)-face is hyperface).

Let $d \ge 1$, and $i \in [0..d]$. Let $[\mathbf{v}]^d$ be $d+1$ points in $\mathbb{R}^d$.

Then, we have $[\mathbf{v}_{\theta_i^{d-1}}]^{d-1,d} = (\mathbf{v}_j)_{j \in [0..d] \setminus \{i\}}$.

Moreover, if $[\mathbf{v}]^d$ is affinely independent, then we have $H_i[\mathbf{v}]^d = F_{\theta_i^{d-1}}[\mathbf{v}]^d$ and $H_i[\mathbf{v}]^d = F_{\theta_i^{d-1}}[\mathbf{v}]^d$.

**Proof.**

Direct consequence of Lemma 1368 (jump enumeration, with $n \stackrel{\text{def.}}{=} d-1$, injectivity and image of $\theta_i^{d-1}$), Definition 1568 (l-face affine space, with $l \stackrel{\text{def.}}{=} d-1$ and $\pi^l \stackrel{\text{def.}}{=} \theta_i^{d-1}$), Lemma 1563 (equivalent definition of face hyperplane), Definition 1566 (hyperface), and Definition 1572 (l-face). $\square$

### 9.5.3 Geometric l-face mappings

*Remark 1577.* In this section, we assume that $l \ge 1$ to avoid treating polynomials over 0-dimensional affine spaces.

**Definition 1578 (geometric l-face mapping).** Let $d \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ points in $\mathbb{R}^d$. Let $l \in [1..d]$. Let $\pi^l$ be an injective map from $[0..l]$ to $[0..d]$. The geometric l-face mapping associated with $[\mathbf{v}]^d$ and the l-face $F_{\pi^l}[\mathbf{v}]^d$ is denoted $\phi_{\pi^l}[\mathbf{v}]^d$, and is defined by

$$(9.65) \qquad \forall \hat{\mathbf{x}} \in \mathbb{R}^l, \quad \phi_{\pi^l}[\mathbf{v}]^d(\hat{\mathbf{x}}) \stackrel{\text{def.}}{=} \sum_{j=0}^{l} \hat{\mathcal{L}}_{j}^{1,l}(\hat{\mathbf{x}}) \mathbf{v}_{\pi^l(j)} \in \mathbb{R}^d.$$

**Lemma 1579 (geometric d-face mapping is geometric mapping).**

Let $d \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ points in $\mathbb{R}^d$. Then, we have $\phi_{\text{Id}_{\mathbb{R}^d}}[\mathbf{v}]^d = \varphi_{\text{geo}}[\mathbf{v}]^d$.

**Proof.** Direct consequence of **injectivity of the identity map**, Definition 1578 (geometric l-face mapping), and Definition 1547 (geometric mapping). $\square$

*Remark 1580.* The next lemma (on $\phi_{\pi^l}[\mathbf{v}]^d$) is an extension of Lemma 1550 (on $\varphi_{\text{geo}}[\mathbf{v}]^d$).
**Lemma 1581 (properties of geometric l-face mapping).** Let $d \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ points in $\mathbb{R}^d$. Let $l \in [1..d]$. Let $\pi^l$ an injective map from $[0..l]$ to $[0..d]$.

Let $B_{\pi^l}^{[\mathbf{v}]^d} \stackrel{\text{def.}}{=} (\mathbf{v}_{\pi^l(1)} - \mathbf{v}_{\pi^l(0)} \ \mathbf{v}_{\pi^l(2)} - \mathbf{v}_{\pi^l(0)} \ \dots \ \mathbf{v}_{\pi^l(l)} - \mathbf{v}_{\pi^l(0)})$ in $\mathcal{M}_{d,l}$.

Then, $\phi_{\pi^l}^{[\mathbf{v}]^d}$ belongs to $\mathcal{A}\mathcal{f}(\mathbb{R}^l, \mathbb{R}^d)$, and we have

$$ (9.66) \quad \forall j \in [0..l], \quad \forall \hat{\mathbf{x}} \in \mathbb{R}^l, \quad \phi_{\pi^l}^{[\mathbf{v}]^d}(\hat{\mathbf{x}}) = \mathbf{v}_{\pi^l(j)} + \sum_{k=0, k \neq j}^{l} \hat{\mathcal{L}}_{\pi^l}^{1,l}(\hat{\mathbf{x}})(\mathbf{v}_{\pi^l(k)} - \mathbf{v}_{\pi^l(j)}), $$

$$ (9.67) \quad \forall \hat{\mathbf{x}} \in \mathbb{R}^l, \quad \phi_{\pi^l}^{[\mathbf{v}]^d}(\hat{\mathbf{x}}) = \mathbf{v}_{\pi^l(0)} + \sum_{j=1}^{l} \hat{x}_j(\mathbf{v}_{\pi^l(j)} - \mathbf{v}_{\pi^l(0)}) = \mathbf{v}_{\pi^l(0)} + B_{\pi^l}^{[\mathbf{v}]^d} \hat{\mathbf{x}}, $$

$$ (9.68) \quad \forall j \in [0..l], \quad \phi_{\pi^l}^{[\mathbf{v}]^d}(\hat{\mathbf{v}}_j^l) = \mathbf{v}_{\pi^l(j)} \quad (\text{where } \hat{\mathbf{v}}_j^l \text{ is a reference vertex in } \mathbb{R}^{l-1}), $$

$$ (9.69) \quad \phi_{\pi^l}^{[\mathbf{v}]^d}(\mathbb{R}^l) = \mathcal{F}_{\pi^l}^{[\mathbf{v}]^d}, $$

$$ (9.70) \quad \phi_{\pi^l}^{[\mathbf{v}]^d}(\hat{\mathbf{K}}^l) = F_{\pi^l}^{[\mathbf{v}]^d}. $$

Moreover, if $[\mathbf{v}]^d$ is affinely independent, then $\phi_{\pi^l}^{[\mathbf{v}]^d}$ is bijective from $\mathbb{R}^l$ onto $\mathcal{F}_{\pi^l}^{[\mathbf{v}]^d}$, its inverse $(\phi_{\pi^l}^{[\mathbf{v}]^d})^{-1}$ from $\mathcal{F}_{\pi^l}^{[\mathbf{v}]^d}$ onto $\mathbb{R}^l$ is affine, and we have

$$ (9.71) \quad \forall j \in [0..l], \quad (\phi_{\pi^l}^{[\mathbf{v}]^d})^{-1}(\mathbf{v}_{\pi^l(j)}) = \hat{\mathbf{v}}_j^l. $$

**Proof. Identities (9.66) and (9.67),** $\phi_{\pi^l}^{[\mathbf{v}]^d}$ **affine.** Let $j \in [0..l]$. Then, from Definition 1578 (geometric l-face mapping), Lemma 1543 (reference Lagrange polynomials is basis of $\mathbb{P}_1^d$, (9.33) with $d \stackrel{\text{def.}}{=} l$, extracting $j$), and **ring properties of $\mathbb{R}$**, identity (9.66) holds. Thus, from (9.66) (with $j \stackrel{\text{def.}}{=} 0$), Definition 1541 (reference Lagrange polynomials of $\mathbb{P}_1^d$, (9.30)), **the columnwise rule for matrix-vector product, commutative ring properties of $\mathbb{R}$**, and Definition 1388 (affine map, with $\phi \stackrel{\text{def.}}{=} B_{\pi^l}^{[\mathbf{v}]^d}$ and $\mathbf{y}_0 \stackrel{\text{def.}}{=} \mathbf{v}_{\pi^l(0)}$), identities (9.67) hold, and $\phi_{\pi^l}^{[\mathbf{v}]^d}$ is affine.

**Identity (9.68).** Direct consequence of Definition 1578 (geometric l-face mapping), and Lemma 1543 (reference Lagrange polynomials is basis of $\mathbb{P}_1^d$, (9.32)).

**Identity (9.69).** $\phi_{\pi^l}^{[\mathbf{v}]^d}(\mathbb{R}^l) \subset \mathcal{F}_{\pi^l}^{[\mathbf{v}]^d}$.

Direct consequence of Definition 1578 (geometric l-face mapping), Lemma 1543 (reference Lagrange polynomials is basis of $\mathbb{P}_1^d$, (9.33)), and Definition 1568 (l-face affine space).

$\mathcal{F}_{\pi^l}^{[\mathbf{v}]^d} \subset \phi_{\pi^l}^{[\mathbf{v}]^d}(\mathbb{R}^l)$. From Definition 1568 (l-face affine space), let $\mathbf{x} = \sum_{j=0}^l \mu_j \mathbf{v}_{\pi^l(j)} \in \mathcal{F}_{\pi^l}^{[\mathbf{v}]^d}$ with $\sum_{j=0}^l \mu_j = 1$. Let $\hat{\mathbf{x}} \stackrel{\text{def.}}{=} (\mu_j)_{j \in [1..l]} \in \mathbb{R}^l$. Then, from Definition 1541 (reference Lagrange polynomials of $\mathbb{P}_1^d$), and Definition 1578 (geometric l-face mapping), we have for all $j \in [1..l]$, $\hat{\mathcal{L}}_{\pi^l}^{1,l}(\hat{\mathbf{x}}) = \mu_j$ and $\hat{\mathcal{L}}_{0}^{1,l}(\hat{\mathbf{x}}) = 1 - \sum_{j=1}^l \mu_j = \mu_0$, thus $\mathbf{x} = \phi_{\pi^l}^{[\mathbf{v}]^d}(\hat{\mathbf{x}}) \in \phi_{\pi^l}^{[\mathbf{v}]^d}(\mathbb{R}^l)$. Therefore, we have the equality.

**Identity (9.70).** $\phi_{\pi^l}^{[\mathbf{v}]^d}(\hat{\mathbf{K}}^l) \subset \mathcal{F}_{\pi^l}^{[\mathbf{v}]^d}$. Direct consequence of Definition 1578 (geometric l-face mapping), Definition 1437 (reference simplex), Definition 1541 (reference Lagrange polynomials of $\mathbb{P}_1^d$, thus for all $\hat{\mathbf{x}} \in \hat{\mathbf{K}}^l$, for all $i \in [0..l]$, $\hat{\mathcal{L}}_i^{1,l}(\hat{\mathbf{x}}) \ge 0$), and Lemma 1543 (reference Lagrange polynomials is basis of $\mathbb{P}_1^d$, (9.33)), and Definition 1572 (l-face).

$\mathcal{F}_{\pi^l}^{[\mathbf{v}]^d} \subset \phi_{\pi^l}^{[\mathbf{v}]^d}(\hat{\mathbf{K}}^l)$. From Definition 1572 (l-face), let $\mathbf{x} = \sum_{j=0}^l \mu_j \mathbf{v}_{\pi^l(j)} \in \mathcal{F}_{\pi^l}^{[\mathbf{v}]^d}$ with for all $j \in [0..l]$, $\mu_j \ge 0$ and $\sum_{j=0}^l \mu_j = 1$. Let $\hat{\mathbf{x}} \stackrel{\text{def.}}{=} (\mu_j)_{j \in [1..l]} \in \mathbb{R}^l$. Then, from Definition 1541 (reference Lagrange polynomials of $\mathbb{P}_1^d$), Definition 1437 (reference simplex), and Definition 1578
(geometric l-face mapping), we have

$$ \forall j \in [1..l], \hat{x}_j = \hat{\mathcal{L}}_j^{1,l}(\hat{\mathbf{x}}) = \mu_j \ge 0 \quad \text{and} \quad 1 - \sum_{j=1}^{l} \hat{x}_j = \hat{\mathcal{L}}_0^{1,l}(\hat{\mathbf{x}}) = 1 - \sum_{j=1}^{l} \mu_j = \mu_0 \ge 0, $$

thus $\hat{\mathbf{x}} \in \hat{\mathcal{K}}^l$ and $\mathbf{x} = \phi_{\pi^l}^{[\mathbf{v}]}(\hat{\mathbf{x}}) \in \phi_{\pi^l}^{[\mathbf{v}]}(\hat{\mathcal{K}}^l)$. Therefore, we have the equality.

**Bijection.** Direct consequence of (9.67), Lemma 1406 (injective affine map is zero linear kernel), Definition 101 (kernel), Lemma 1414 (affinely independent family is closed by sub-family, with $j \stackrel{\text{def.}}{=} 0$, thus $(\mathbf{v}_{\pi^l(i)} - \mathbf{v}_{\pi^l(0)})_{i \in [1..l]}$ is free), (9.69), and **the definition of bijectivity**.

$$(\phi_{\pi^l}^{[\mathbf{v}]^d})^{-1} \text{ affine.} \text{ Direct consequence of (9.67), Lemma 1409 (inverse of affine submap is affine submap, with } E = E' = \mathcal{E}' \stackrel{\text{def.}}{=} \mathbb{R}^l \text{ } (\mathbf{x}_0 \stackrel{\text{def.}}{=} \mathbf{0}), F \stackrel{\text{def.}}{=} \mathbb{R}^d, \mathcal{F}' \stackrel{\text{def.}}{=} \mathcal{F}_{\pi^l}^{[\mathbf{v}]^d}, \text{ and } \mathbf{y}'_0 \stackrel{\text{def.}}{=} \mathbf{v}_{\pi^l(0)} \in \mathcal{F}_{\pi^l}^{[\mathbf{v}]^d}), \text{ and Lemma 1569 (equivalent definition of l-face affine space).}$$

**Equation (9.71).** Direct consequence of Equation (9.68). □

**Lemma 1582 (geometric l-face mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$).**

Let $d \ge 1$. Let $k \in \mathbb{N}$. Let $[\mathbf{v}]^d$ be $d+1$ points in $\mathbb{R}^d$. Let $l \in [1..d]$. Let $\pi^l$ an injective map from $[0..l]$ to $[0..d]$. Let $p \in \mathbb{P}_k^d$. Then, we have $p \circ \phi_{\pi^l}^{[\mathbf{v}]^d} \in \mathbb{P}_k^l$.

**Proof.** Direct consequence of Lemma 1581 (properties of geometric l-face mapping, thus $\phi_{\pi^l}^{[\mathbf{v}]^d}$ belongs to $\mathcal{Aff}(\mathbb{R}^l, \mathbb{R}^d)$), and Lemma 1540 (affine mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$). □

**Lemma 1583 (geometric mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$).**

Let $d \ge 1$. Let $k \in \mathbb{N}$. Let $[\mathbf{v}]^d$ be $d+1$ points in $\mathbb{R}^d$. Let $p \in \mathbb{P}_k^d$. Then, we have $p \circ \phi_{\text{geo}}^{[\mathbf{v}]^d} \in \mathbb{P}_k^d$. Moreover, if $[\mathbf{v}]^d$ is affinely independent, then $p \circ (\phi_{\text{geo}}^{[\mathbf{v}]^d})^{-1}$ also belongs to $\mathbb{P}_k^d$.

**Proof.** Direct consequence of Lemma 1579 (geometric d-face mapping is geometric mapping), Lemma 1582 (geometric l-face mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$, with $l \stackrel{\text{def.}}{=} d$ and $\pi^l \stackrel{\text{def.}}{=} \text{Id}_{\mathbb{R}^d}$), Lemma 1550 (properties of geometric mapping, thus $(\phi_{\text{geo}}^{[\mathbf{v}]^d})^{-1}$ is affine), and Lemma 1540 (affine mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$, with $l \stackrel{\text{def.}}{=} d$). □

### 9.5.4 Geometric hyperface mapping

**Lemma 1584 (geometric hyperface mapping).** Let $d \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ points in $\mathbb{R}^d$.

Let $i \in [0..d]$. Let $\hat{\mathbf{x}} \in \mathbb{R}^{d-1}$. Then, we have $\phi_{\theta^{d-1}}^{[\mathbf{v}]} \in \mathcal{Aff}(\mathbb{R}^{d-1}, \mathbb{R}^d)$ and

$$ (9.72) \qquad (i=0) \qquad \phi_{\theta_0^{d-1}}^{[\mathbf{v}]^d}(\hat{\mathbf{x}}) = \mathbf{v}_1 + \sum_{j=1}^{d-1} \hat{x}_j(\mathbf{v}_{j+1} - \mathbf{v}_1), $$

$$ (9.73) \qquad (i \in [1..d-1]) \qquad \phi_{\theta_i^{d-1}}^{[\mathbf{v}]^d}(\hat{\mathbf{x}}) = \mathbf{v}_0 + \sum_{j=1}^{i-1} \hat{x}_j(\mathbf{v}_j - \mathbf{v}_0) + \sum_{j=i}^{d-1} \hat{x}_j(\mathbf{v}_{j+1} - \mathbf{v}_0), $$

$$ (9.74) \qquad \forall j \in [0..d] \qquad \phi_{\theta_i^{d-1}}^{[\mathbf{v}]^d}(\hat{\mathbf{v}}_j^{d-1}) = \begin{cases} \mathbf{v}_j & \text{if } j < i, \\ \mathbf{v}_{j+1} & \text{if } j \ge i. \end{cases} $$

Moreover, if $[\mathbf{v}]^d$ is affinely independent, then $\phi_{\theta^{d-1}}^{[\mathbf{v}]^d}$ is bijective from $\mathbb{R}^{d-1}$ onto $\mathcal{H}_i^{[\mathbf{v}]^d}$, and its inverse $(\phi_{\theta^{d-1}}^{[\mathbf{v}]^d})^{-1}$ from $\mathcal{H}_i^{[\mathbf{v}]^d}$ onto $\mathbb{R}^{d-1}$ is affine.
<figure><img src="image_30.png" /><figcaption>Figure 9.6: Geometric hyperface mapping in the case $d = k = 3$ (see Lemma 1584).</figcaption></figure>

The reference triangle $\hat{K}_2$ is mapped onto $H_0^2[\mathbf{v}]^d$, the 0-th face opposite vertex $\mathbf{v}_0$ that is depicted in blue. The correspondence between the reference nodes in the triangle ($d=2$) and the face nodes of the tetrahedron ($d=3$) is illustrated by the colors: we have $\phi_{\theta_0^{d-1}}^{[\mathbf{v}]^d}(\hat{a}_{i,j}) = a_{(3-(i+j),i,j)}$, for all $(i,j) \in A_2^3$ (see Lemma 1607).

*Proof.* Direct consequence of Lemma 1576 ((d-1)-face is hyperface), and Lemma 1581 (properties of geometric l-face mapping). $\square$

**Lemma 1585 (hyperface geometric mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^{d-1}$).** Let $d \ge 2$. Let $k \in \mathbb{N}$.
Let $[\mathbf{v}]^d$ be $d+1$ points in $\mathbb{R}^d$. Let $i \in [0..d]$. Let $p \in \mathbb{P}_k^d$. Then, we have $p \circ \phi_{\theta_i^{d-1}}^{[\mathbf{v}]^d} \in \mathbb{P}_k^{d-1}$.

*Proof.* Direct consequence of Lemma 1576 ((d-1)-face is hyperface), and Lemma 1582 (geometric l-face mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$, with $l \stackrel{\text{def.}}{=} d-1$ and $\pi^l \stackrel{\text{def.}}{=} \theta_i^{d-1}$). $\square$
<figure><img src="image_31.png" /><figcaption>Figure 9.7: Geometric transformation $\phi_{c_0^d[\mathbf{v}]^d}^{}$ in the case $d = k = 3$, with the circular permutation $c_0^d$ that maps $(0, 1, 2, 3)$ to $(1, 2, 3, 0)$, see Lemma 1366.</figcaption></figure>

The reference simplex $\hat{K}^d$ is mapped onto the current simplex $K[\mathbf{v}]^d$, see Lemma 1586. The reference vertices are mapped onto the current vertices with a change of indices: for all $j \in [0..d]$, we have $\phi_{c_0^d[\mathbf{v}]^d}(\hat{v}_j) = \mathbf{v}_{c_0^d(j)}$, see Lemma 1581. Note that as $c_0^d(d) = 0$, the reference hyperplane $\hat{H}_d^d$ (containing the face $\hat{H}_d^0$ opposite vertex $\hat{v}_d$) is mapped onto $\mathcal{H}_0^{[\mathbf{v}]^d}$ (containing the face $\mathcal{H}_0^{[\mathbf{v}]^d}$ opposite vertex $\mathbf{v}_0$). In this figure, only the visible nodes are depicted.

### 9.5.5 Geometric mapping with permutation

**Lemma 1586 (geometric mapping with permutation).** Let $d \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$. Let $\pi^d$ be an injective mapping from $[0..d]$ to $[0..d]$. Let $\hat{\mathbf{x}} \in \mathbb{R}^d$. Then, $\pi^d$ is a bijective permutation of $[0..d]$, $\phi_{\pi^d}^{[\mathbf{v}]^d} \in \mathcal{A}ff(\mathbb{R}^d, \mathbb{R}^d)$ is bijective, and we have

$$ (9.75) \qquad \phi_{\pi^d}^{[\mathbf{v}]^d}(\hat{\mathbf{x}}) = \sum_{i=0}^{d} \hat{\mathcal{L}}_{(\pi^d)^{-1}(i)}^{1,d}(\hat{\mathbf{x}}) \mathbf{v}_i $$

$$ (9.76) \qquad \forall i \in [0..d], \quad \phi_{\pi^d}^{[\mathbf{v}]^d}(\hat{\mathbf{x}}) = \mathbf{v}_i + \sum_{k=0, k \neq i}^{d} \hat{\mathcal{L}}_{(\pi^d)^{-1}(k)}^{1,d}(\hat{\mathbf{x}})(\mathbf{v}_k - \mathbf{v}_i), $$

$$ (9.77) \qquad \forall i \in [0..d], \quad \lambda_i^{[\mathbf{v}]^d} = \mathcal{L}_i^{[\mathbf{v}]^d} = \hat{\mathcal{L}}_{(\pi^d)^{-1}(i)}^{1,d} \circ (\phi_{\pi^d}^{[\mathbf{v}]^d})^{-1}, $$

$$ (9.78) \qquad \forall j \in [0..d], \quad \phi_{\pi^d}^{[\mathbf{v}]^d}(\hat{\mathcal{H}}_j^d) = \mathcal{H}_{\pi^d(j)}^{[\mathbf{v}]^d}, $$

$$ (9.79) \qquad \phi_{\pi^d}^{[\mathbf{v}]^d}(\hat{K}^d) = K^{[\mathbf{v}]^d}. $$

*Proof. Bijections, $\phi_{\pi^d}^{[\mathbf{v}]^d}$ affine, identities (9.75), (9.76). Direct consequence of Lemma 1574 (d-face is simplex), Lemma 1581 (properties of geometric l-face mapping, with $l \stackrel{\text{def.}}{=} d$), Lemma 1570 (d-face affine space is full space), and Definition 1578 (geometric l-face mapping).*

**Identities (9.77).** Let $\mathbf{x} \in \mathbb{R}^d$. Let $\hat{\mathbf{y}} \stackrel{\text{def.}}{=} (\phi_{\pi^d}^{[\mathbf{v}]^d})^{-1}(\mathbf{x}) \in \mathbb{R}^d$, i.e. such that $\phi_{\pi^d}^{[\mathbf{v}]^d}(\hat{\mathbf{y}}) = \mathbf{x}$. From Lemma 1559 (barycentric coordinate, decomposition, then uniqueness), (9.75), Lemma 1543 (reference Lagrange polynomials is basis of $\mathbb{P}_1^d$, (9.33)), and Lemma 1560 (Lagrange polynomials
of $\mathbb{P}_1^d$ are barycentric coordinate), we have

$$ \mathbf{x} = \sum_{i=0}^{d} \lambda_{i}^{[\mathbf{v}]^{d}}(\mathbf{x}) \mathbf{v}_{i} = \sum_{i=0}^{d} \hat{\mathcal{L}}_{(\pi^{d})^{-1}(i)}^{1,d}(\hat{\mathbf{y}}) \mathbf{v}_{i} = \sum_{i=0}^{d} \hat{\mathcal{L}}_{(\pi^{d})^{-1}(i)}^{1,d} \circ (\phi_{\pi^{d}}^{[\mathbf{v}]^{d}})^{-1}(\mathbf{x}) \mathbf{v}_{i} $$

and $\sum_{i=0}^{d} \hat{\mathcal{L}}_{(\pi^{d})^{-1}(i)}^{1,d} = 1$. Thus, for all $i \in [0..d]$, $\hat{\mathcal{L}}_{(\pi^{d})^{-1}(i)}^{1,d} \circ (\phi_{\pi^{d}}^{[\mathbf{v}]^{d}})^{-1} = \lambda_{i}^{[\mathbf{v}]^{d}} = \mathcal{L}_{i}^{[\mathbf{v}]^{d}}$.

**Identity (9.78).** Let $j \in [0..d]$. Let $i \stackrel{\text{def.}}{=} \pi^d(j)$, i.e. such that $(\pi^d)^{-1}(i) = j$. Then, from Lemma 1564 (reference face hyperplane), (9.77) (thus $\hat{\mathcal{L}}_{j}^{1,d} = \lambda_{i}^{[\mathbf{v}]^{d}} \circ \phi_{\pi^{d}}^{[\mathbf{v}]^{d}}$), the definition of bijectivity, Lemma 1370 (image of ker is ker, with $f \stackrel{\text{def.}}{=} \phi_{\pi^{d}}^{[\mathbf{v}]^{d}}$ surjective, and $g \stackrel{\text{def.}}{=} \lambda_{i}^{[\mathbf{v}]^{d}}$), and Lemma 1563 (equivalent definition of face hyperplane), we have

$$ \phi_{\pi^d}^{[\mathbf{v}]^d}(\hat{\mathcal{H}}_j^d) = \phi_{\pi^d}^{[\mathbf{v}]^d}(\ker(\hat{\mathcal{L}}_j^{1,d})) = \ker(\lambda_i^{[\mathbf{v}]^d}) = \mathcal{H}_i^{[\mathbf{v}]^d}. $$

**Identity (9.79).** Direct consequence of Lemma 1581 (properties of geometric l-face mapping, (9.70) with $l \stackrel{\text{def.}}{=} d$), and Lemma 1574 (d-face is simplex). $\square$

## 9.6 Lagrange nodes for $\mathbb{L}_{\mathcal{P}_k^d}$

*Remark 1587.* We recall the distinct notations for vertices $[[\mathbf{v}]^d]$, and for nodes $\{\mathbf{a}\}_{\mathcal{A}_k^d}$ (see Remark 1433), as well as Definition 1401 for isobarycenters.

### 9.6.1 Lagrange nodes for the current element

**Definition 1588 (Lagrange nodes of $\mathbb{P}_k^d$).** Let $d \ge 1$. Let $k \in \mathbb{N}$. Let $[[\mathbf{v}]^d$ be $d+1$ points in $\mathbb{R}^d$. The Lagrange nodes of $\mathbb{P}_k^d$ are denoted $\{\mathbf{a}\}_{\mathcal{A}_k^d} = (\mathbf{a}_{\alpha})_{\alpha \in \mathcal{A}_k^d}$, and are defined by

$$ (9.80) \qquad (k=0) \qquad \mathbf{a}_0 \stackrel{\text{def.}}{=} \mathbf{g}^{[\mathbf{v}]^d} \left( = \frac{1}{d+1} \sum_{i=0}^{d} \mathbf{v}_i \right), $$

$$ (9.81) \qquad (k \ge 1) \qquad \forall \alpha \in \mathcal{A}_k^d, \quad \mathbf{a}_\alpha \stackrel{\text{def.}}{=} \mathbf{v}_0 + \sum_{i=1}^{d} \frac{\alpha_i}{k} (\mathbf{v}_i - \mathbf{v}_0). $$

**Lemma 1589 (Lagrange nodes of $\mathbb{P}_k^d$ for $d=1$ are Lagrange nodes of $\mathbb{P}_k^1$).**

Let $k \in \mathbb{N}$. Let $[[\mathbf{v}]^1 = [[v]^1$ be two points in $\mathbb{R}^1$.

Then, $\{\mathbf{a}\}_{\mathcal{A}_k^d}$ for $d \stackrel{\text{def.}}{=} 1$, and $\{\mathbf{a}\}_{\mathcal{A}_k^d}$ from Definition 1465 coincide.

**Proof.** Direct consequence of Definition 1588 (Lagrange nodes of $\mathbb{P}_k^d$, with $d \stackrel{\text{def.}}{=} 1$), Lemma 1497 (first multi-indices $\mathcal{A}_k^d$, with $d \stackrel{\text{def.}}{=} 1$), and Definition 1465 (Lagrange nodes of $\mathbb{P}_k^1$). $\square$

**Lemma 1590 (number of Lagrange nodes of $\mathbb{P}_k^d$).**

Let $d \ge 1$. Let $k \in \mathbb{N}$. Let $[[\mathbf{v}]^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$. Let $\alpha, \beta \in \mathcal{A}_k^d$.

Then, $\alpha \ne \beta$ implies $\mathbf{a}_{\alpha} \ne \mathbf{a}_{\beta}$. Thus, there are $\binom{k+d}{d}$ distinct Lagrange nodes in $\{\mathbf{a}\}_{\mathcal{A}_k^d}$.

**Proof.** Direct consequence of Definition 1588 (Lagrange nodes of $\mathbb{P}_k^d$), Lemma 1559 (barycentric coordinate, uniqueness in (9.51) with $j \stackrel{\text{def.}}{=} 0$), and Lemma 1498 (cardinal of $\mathcal{A}_k^d$). $\square$
**Lemma 1591 (barycentric coordinates of Lagrange nodes of $\mathbb{P}_k^d$).** Let $d \ge 1$. Let $k \in \mathbb{N}$. Let $[[v]]^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$. Let $\alpha \in \mathcal{A}_k^d$. Let $i \in [0..d]$. Then, the barycentric coordinates of the Lagrange node $\mathbf{a}_\alpha$ are

$$ (9.82) \qquad (k=0) \qquad \lambda_i^{|v|} (\mathbf{a}_0) = \frac{1}{d+1} \in [0,1], $$

$$ (9.83) \qquad (k \ge 1) \qquad \left\{ \begin{array}{ll} (i=0) & \lambda_0^{|v|} (\mathbf{a}_\alpha) = 1 - \frac{|\alpha|}{k} \in [0,1], \\ (i \in [1..d]) & \lambda_i^{|v|} (\mathbf{a}_\alpha) = \frac{\alpha_i}{k} \in [0,1]. \end{array} \right. $$

Moreover, if $k \ge 1$, then the Lagrange node is equivalently defined by

$$ (9.84) \qquad \mathbf{a}_{\alpha} = \left(1 - \frac{|\alpha|}{k}\right) \mathbf{v}_0 + \sum_{i=1}^{d} \frac{\alpha_i}{k} \mathbf{v}_i \quad \text{with} \quad \left(1 - \frac{|\alpha|}{k}\right) + \sum_{i=1}^{d} \frac{\alpha_i}{k} = 1. $$

*Proof.* Direct consequence of Definition 1588 (Lagrange nodes of $\mathbb{P}_k^d$), Definition 1401 (isobarycenter, for $k=0$), Lemma 1559 (barycentric coordinate, (9.50) for $k=0$, (9.50) and (9.51) with $j \stackrel{\text{def.}}{=} 0$ for $k \ge 1$, and uniqueness), Definition 1486 (sets of multi-indices $\mathcal{A}_k^d$ and $\mathcal{C}_k^d$, for $k \ge 1$), Definition 1480 (length of multi-indices, for $k \ge 1$), Lemma 1488 (indices are smaller than maximal length, for $k \ge 1$), and ordered field properties of $\mathbb{R}$ (with $d+1 > 0$ for $k=0$, and $k > 0$ for $k \ge 1$). $\square$

**Lemma 1592 (vertices are Lagrange nodes of $\mathbb{P}_k^d$).**
Let $d \ge 1$. Let $k \ge 1$. Let $[[v]]^d$ be $d+1$ points in $\mathbb{R}^d$. Then, the vertices are nodes of $\{\mathbf{a}\}_{\mathcal{A}_1^d}$,

$$ (9.85) \qquad \mathbf{v}_0 = \mathbf{a}_0 \quad \text{and} \quad \forall i \in [1..d], \quad \mathbf{v}_i = \mathbf{a}_{k\mathbf{e}_i}. $$

*Proof.* Case $i=0$. Direct consequence of Definition 1588 (Lagrange nodes of $\mathbb{P}_k^d$, with $k > 0$).

Case $i \in [1..d]$. From Definition 1365 (canonic families), Definition 1480 (length of multi-indices), Definition 1486 (sets of multi-indices $\mathcal{A}_k^d$ and $\mathcal{C}_k^d$), and Definition 1588 (Lagrange nodes of $\mathbb{P}_k^d$, with $k > 0$), we have $|k\mathbf{e}_i| = k \sum_{j=1}^d \delta_{ij} = k \le k$, thus $k\mathbf{e}_i \in \mathcal{A}_k^d$, and

$$ \mathbf{a}_{k\mathbf{e}_i} = \mathbf{v}_0 + \sum_{j=1}^{d} \frac{k\delta_{ij}}{k}(\mathbf{v}_j - \mathbf{v}_0) = \mathbf{v}_0 + (\mathbf{v}_i - \mathbf{v}_0) = \mathbf{v}_i. $$

Therefore, the identities hold. $\square$

**Lemma 1593 (Lagrange nodes of $\mathbb{P}_1^d$ are vertices).**
Let $d \ge 1$. Let $[[v]]^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$. Then, the nodes of $\{\mathbf{a}\}_{\mathcal{A}_1^d}$ are the vertices,

$$ (9.86) \qquad \mathbf{a}_0 = \mathbf{v}_0 \quad \text{and} \quad \forall i \in [1..d], \quad \mathbf{a}_{\mathbf{e}_i} = \mathbf{v}_i. $$

*Proof.* Direct consequence of Lemma 1497 (first multi-indices $\mathcal{A}_k^d$, with $k=1$), and Lemma 1592 (vertices are Lagrange nodes of $\mathbb{P}_k^d$, with $k=1$). $\square$

### 9.6.2 Sub-vertices and sub-nodes

**Definition 1594 (sub-vertices of Lagrange nodes of $\mathbb{P}_k^d$).**
Let $d \ge 1$. Let $k \ge 1$. Let $[[v]]^d$ be $d+1$ points in $\mathbb{R}^d$. Let $i \in [0..d]$. The sub-vertices of the Lagrange nodes of $\mathbb{P}_k^d$ with respect to $\mathbf{v}_0$ are denoted $[[v]]^d = (\mathbf{v}_i)_{i \in [0..d]}$, and are defined by

$$ (9.87) \qquad (i=0) \qquad \mathbf{v}_0 \stackrel{\text{def.}}{=} \mathbf{v}_0, $$

$$ (9.88) \qquad (i \in [1..d]) \qquad \mathbf{v}_i \stackrel{\text{def.}}{=} \mathbf{a}_{(k-1)\mathbf{e}_i}. $$
**Lemma 1595 (equivalent definition of sub-vertices of Lagrange nodes of $\mathbb{P}_k^d$).**

Let $d \ge 1$. Let $k \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ points in $\mathbb{R}^d$. Let $i \in [0..d]$. Then, we have

$$ (9.89) \qquad \mathbf{v}_i = \frac{1}{k}\mathbf{v}_0 + \frac{k-1}{k}\mathbf{v}_i. $$

*Proof.* Direct consequence of Definition 1594 (sub-vertices of Lagrange nodes of $\mathbb{P}_k^d$), ordered field properties of $\mathbb{R}$ (with $k > 0$), thus $\frac{1}{k} + \frac{k-1}{k} = 1$ (case $i = 0$), $1 - \frac{k-1}{k} = \frac{1}{k}$, and $0 \le k-1 < k$, Definition 1365 (canonic families), Definition 1480 (length of multi-indices, thus $|(k-1)\mathbf{e}_i| = k-1$), Definition 1486 (sets of multi-indices $\mathcal{A}_k^d$ and $C_k^d$, thus $(k-1)\mathbf{e}_i \in \mathcal{A}_k^d$), and Lemma 1591 (barycentric coordinates of Lagrange nodes of $\mathbb{P}_k^d$, (9.84)). $\square$

*Remark 1596.* An illustration of the sub-vertices (see Definition 1594) and of $\mathbb{P}_{k-1}^d$ sub-nodes (see Lemma 1598) in dimension $d = 3$ and for $k = 3$ can be seen in Figure 9.8.

The sub-vertices $[\mathbf{v}]^d$ are $\mathbf{v}_0$ and the nodes of $\mathbb{P}_k^d$ that are the closest to the vertices along the axes. The $\mathbb{P}_{k-1}^d$ sub-nodes are the nodes of $\mathbb{P}_k^d$, except the nodes indexed by $\alpha \in C_k^d$ (that lie in the face $H_0^d[\mathbf{v}]^d$). These sub-vertices and sub-nodes are used in the step 2 of the proof of the unisolvence Lemma 1625.

<figure><img src="image_32.png" /><figcaption>Figure 9.8: Sub-vertices $[\mathbf{v}]^d$ with respect to $\mathbf{v}_0$ and Lagrange $\mathbb{P}_{k-1}^d$ sub-nodes $\{\mathbf{a}\}_{\mathcal{A}_{k-1}^d}$, in the case $d = k = 3$.</figcaption></figure>

The Lagrange $\mathbb{P}_k^d$ nodes $\{\mathbf{a}\}_{\mathcal{A}_k^d}$ are set in the tetrahedron defined by the vertices $[\mathbf{v}]^3 = (\mathbf{v}_0, \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3)$.

The sub-vertices are $[\mathbf{v}]^3 \stackrel{\text{def.}}{=} (\mathbf{v}_0, \mathbf{a}_{2e_1}, \mathbf{a}_{2e_2}, \mathbf{a}_{2e_3})$, see Definition 1594 and Remark 1596.

The $\mathbb{P}_{k-1}^d$ sub-nodes $\{\mathbf{a}\}_{\mathcal{A}_2^3}$ are defined with respect to the tetrahedron whose vertices are $[\mathbf{v}]^3$. These are the nodes $\{\mathbf{a}\}_{\mathcal{A}_3^3}$, except the (small blue) nodes corresponding to $k = 3$ (with indices in $C_3^3$), see Lemma 1598.

Thus, geometrically, passing from $\mathbb{P}_k^d$ to $\mathbb{P}_{k-1}^d$ amounts to remove the face $H_0^d[\mathbf{v}]^d$ from the tetrahedron defined by $[\mathbf{v}]^d$.

**Lemma 1597 (sub-vertices are affinely independent).**

Let $d \ge 1$. Let $k \ge 2$. Let $[\mathbf{v}]^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$.

Then, the sub-vertices $[\mathbf{v}]^d$ are affinely independent.

*Proof.* Let $(\mu_i)_{i \in [1..d]}$ such that $\sum_{i=1}^d \mu_i(\mathbf{v}_i - \mathbf{v}_0) = 0$. Then, from Lemma 1595 (equivalent definition of sub-vertices of Lagrange nodes of $\mathbb{P}_k^d$), and field properties of $\mathbb{R}$ (with $k \ne 0$),

$$ \sum_{i=1}^{d} \mu_i (\mathbf{v}_i - \mathbf{v}_0) = \sum_{i=1}^{d} \mu_i \left( \frac{k-1}{k} \mathbf{v}_i + \left( \frac{1}{k} - 1 \right) \mathbf{v}_0 \right) = \sum_{i=1}^{d} \mu_i \frac{k-1}{k} (\mathbf{v}_i - \mathbf{v}_0) = 0. $$
Thus, from Definition 1411 (affinely independent family), the zero-product property in $\mathbb{R}$ (with $\frac{k-1}{k} \neq 0$), and the definition of freedom, we have $\mu_i = 0$ for all $i \in [1..d]$, thus $(\mathbf{v}_i - \mathbf{v}_0)_{i \in [1..d]}$ is a free family, and $[\mathbf{v}]^d$ is affinely independent. $\square$

**Lemma 1598** ($\mathbb{P}_{k-1}^d$ sub-nodes of sub-vertices are some nodes of $\mathbb{P}_{k-1}^d$).

Let $d \ge 1$. Let $k \ge 2$. Let $[\mathbf{v}]^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$. The Lagrange nodes of $\mathbb{P}_{k-1}^d$ with respect to the sub-vertices $[\mathbf{v}]^d$ are called $\mathbb{P}_{k-1}^d$ sub-nodes and denoted as $\{\mathbf{a}\}_{\mathbb{P}_{k-1}^d}^d$.

Then, these sub-nodes are some of the nodes $\{\mathbf{a}\}_{\mathbb{P}_{k-1}^d}^d$ of $\mathbb{P}_k^d$ (defined with respect to $[\mathbf{v}]^d$),

$$ (9.90) \qquad \forall \alpha \in \mathcal{A}_{k-1}^d, \quad \mathbf{a}_{\alpha} = \mathbf{a}_{\alpha}. $$

*Proof.* Let $\alpha \in \mathcal{A}_{k-1}^d$. Then, from Lemma 1597 (sub-vertices are affinely independent), Lemma 1591 (barycentric coordinates of Lagrange nodes of $\mathbb{P}_k^d$, (9.84), first for $\{\mathbf{a}\}_{\mathbb{P}_{k-1}^d}^d$ with $k-1 \ge 1$ and $[\mathbf{v}]^d$, then for $\{\mathbf{a}\}_{\mathbb{P}_{k-1}^d}^d$ with $k \ge 1$ and $[\mathbf{v}]^d$), Lemma 1595 (equivalent definition of sub-vertices of Lagrange nodes of $\mathbb{P}_k^d$), field properties of $\mathbb{R}$ (with $k \ge 2$, thus $k-1 \neq 0$), Definition 1480 (length of multi-indices), and Lemma 1496 ($C_k^d$ are layers of $\mathcal{A}_k^d$, thus $\alpha \in \mathcal{A}_k^d$), we have

$$ \begin{aligned} \mathbf{a}_{\alpha} &= \left(1 - \frac{|\alpha|}{k-1}\right) \mathbf{v}_0 + \sum_{i=1}^{d} \frac{\alpha_i}{k-1} \mathbf{v}_i = \left(1 - \frac{|\alpha|}{k-1}\right) \mathbf{v}_0 + \sum_{i=1}^{d} \frac{\alpha_i}{k-1} \left(\frac{1}{k} \mathbf{v}_0 + \frac{k-1}{k} \mathbf{v}_i\right) \\ &= \left(1 - \frac{|\alpha|}{k-1} + \frac{|\alpha|}{(k-1)k}\right) \mathbf{v}_0 + \sum_{i=1}^{d} \frac{\alpha_i}{k} \mathbf{v}_i = \left(1 - \frac{|\alpha|}{k}\right) \mathbf{v}_0 + \sum_{i=1}^{d} \frac{\alpha_i}{k} \mathbf{v}_i = \mathbf{a}_{\alpha}. \end{aligned} $$

$\square$

### 9.6.3 Lagrange nodes for the reference element

**Lemma 1599** (reference Lagrange nodes of $\mathbb{P}_k^d$).

Let $d \ge 1$. Let $k \in \mathbb{N}$.

The reference Lagrange nodes of $\mathbb{P}_k^d$ are denoted $\{\hat{\mathbf{a}}\}_{\mathbb{P}_k^d}^d$, and are defined as the Lagrange nodes of $\mathbb{P}_k^d$ for the reference vertices. Let $\alpha \in \mathcal{A}_k^d$. Then, we have

$$ (9.91) \qquad (k=0) \qquad \hat{\mathbf{a}}_0 = \hat{\mathbf{g}}^d, $$

$$ (9.92) \qquad (k \ge 1) \qquad \begin{aligned} \hat{\mathbf{a}}_{\alpha} &= \hat{\mathbf{v}}_0 + \sum_{i=1}^{d} \frac{\alpha_i}{k} (\hat{\mathbf{v}}_i - \hat{\mathbf{v}}_0) = \sum_{i=1}^{d} \frac{\alpha_i}{k} \mathbf{e}_i \\ &= \left(1 - \frac{|\alpha|}{k}\right) \hat{\mathbf{v}}_0 + \sum_{i=1}^{d} \frac{\alpha_i}{k} \hat{\mathbf{v}}_i \quad \text{with} \quad \left(1 - \frac{|\alpha|}{k}\right) + \sum_{i=1}^{d} \frac{\alpha_i}{k} = 1. \end{aligned} $$

*Proof.* Direct consequence of Definition 1588 (Lagrange nodes of $\mathbb{P}_k^d$), Lemma 1435 (reference isobarycenter, for $k=0$), Lemma 1591 (barycentric coordinates of Lagrange nodes of $\mathbb{P}_k^d$, (9.84) for $k \ge 1$), and Definition 1434 (family of reference points, for $k \ge 1$). $\square$

**Lemma 1600** (reference Lagrange nodes of $\mathbb{P}_k^d$ for $d=1$ are reference Lagrange nodes of $\mathbb{P}_k^1$).

Let $k \in \mathbb{N}$. Then, $\{\hat{\mathbf{a}}\}_{\mathbb{P}_k^d}^d$ for $d \stackrel{\text{def.}}{=} 1$ and $\{\hat{\mathbf{a}}\}_{\mathbb{P}_k^1}^d$ from Definition 1453 coincide.

*Proof.* Direct consequence of Lemma 1599 (reference Lagrange nodes of $\mathbb{P}_k^d$, with $d \stackrel{\text{def.}}{=} 1$), Definition 1365 (canonic families, $\mathbf{e}_i$), Lemma 1497 (first multi-indices $\mathcal{A}_k^d$, with $d \stackrel{\text{def.}}{=} 1$), and Definition 1453 (reference Lagrange nodes of $\mathbb{P}_k^1$). $\square$
**Lemma 1601 (equivalent definition of reference Lagrange nodes of $\mathbb{P}_k^d$).**
Let $d \ge 1$. Let $k \in \mathbb{N}$. Let $\alpha \in \mathcal{A}_k^d$. Let $i \in [1..d]$. Then, we have

$$ (9.93) \qquad (k = 0) \qquad (\hat{\mathbf{a}}_0)_i = \frac{1}{d+1}, $$

$$ (9.94) \qquad (k \ge 1) \qquad (\hat{\mathbf{a}}_{\alpha})_i = \frac{\alpha_i}{k}. $$

*Proof.* Direct consequence of Lemma 1599 (reference Lagrange nodes of $\mathbb{P}_k^d$), Lemma 1435 (reference isobarycenter, for $k=0$), and Definition 1365 (canonic families, for $k \ge 1$). □

**Lemma 1602 (number of reference Lagrange nodes of $\mathbb{P}_k^d$).**
Let $d \ge 1$. Let $k \in \mathbb{N}$.
Then, there are $\binom{k+d}{d}$ distinct reference Lagrange nodes $\{\hat{\mathbf{a}}_j\}_{j \in \mathcal{A}_k^d}$.

*Proof.* Direct consequence of Lemma 1599 (reference Lagrange nodes of $\mathbb{P}_k^d$), Lemma 1436 (family of reference points is affinely independent), Lemma 1559 (barycentric coordinate, uniqueness in (9.51) with $j$ def. be 0 and $\|\hat{\mathbf{v}}\|^d$), and Lemma 1498 (cardinal of $\mathcal{A}_k^d$). □

*Remark 1603.* In the next lemma, the proof could also have been done using Lemma 1400.

**Lemma 1604 (Lagrange nodes of $\mathbb{P}_k^d$ are image of reference).**
Let $d \ge 1$. Let $k \in \mathbb{N}$.
Let $\|\mathbf{v}\|^d$ be $d+1$ points in $\mathbb{R}^d$. Let $\alpha \in \mathcal{A}_k^d$. Then, we have $\mathbf{a}_\alpha = \varphi_{\text{geo}}^{\|\mathbf{v}\|^d}(\hat{\mathbf{a}}_\alpha)$.

*Proof.* Direct consequence of Lemma 1550 (properties of geometric mapping, (9.38)), Lemma 1601 (equivalent definition of reference Lagrange nodes of $\mathbb{P}_k^d$), field properties of $\mathbb{R}$ (with $d+1 \ne 0$) for $k=0$, and Definition 1588 (Lagrange nodes of $\mathbb{P}_k^d$). □

### 9.6.4 Lagrange nodes and face hyperplanes

**Lemma 1605 (face hyperplanes of Lagrange nodes of $\mathbb{P}_k^d$).**
Let $d \ge 1$. Let $k \ge 1$.
Let $\|\mathbf{v}\|^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$. Let $\alpha \in \mathcal{A}_k^d$. Let $i \in [0..d]$. Then, we have

$$ (9.95) \qquad (i = 0) \qquad \mathbf{a}_{\alpha} \in \mathcal{H}_{0}^{\|\mathbf{v}\|^d} \iff |\alpha| = k \iff \alpha \in \mathcal{C}_{k}^{d}, $$

$$ (9.96) \qquad (i \in [1..d]) \qquad \mathbf{a}_{\alpha} \in \mathcal{H}_{i}^{\|\mathbf{v}\|^d} \iff \alpha_i = 0 \iff \alpha \in \mathcal{A}_{k,i}^d. $$

Thus, we have $\text{card}\{\mathbf{a}_{\alpha} \in \mathcal{H}_{i}^{\|\mathbf{v}\|^d} \mid \alpha \in \mathcal{A}_k^d\} = \binom{k+d-1}{d-1}$.

*Proof. Equivalences.* Direct consequence of Lemma 1563 (equivalent definition of face hyperplane, (9.57)), Lemma 1591 (barycentric coordinates of Lagrange nodes of $\mathbb{P}_k^d$, (9.83)), field properties of $\mathbb{R}$ (with $k > 0$), Definition 1486 (sets of multi-indices $\mathcal{A}_k^d$ and $\mathcal{C}_k^d$), Lemma 1501 (cardinal of $\mathcal{A}_{k,i}^d$ and $\mathcal{A}_{k,i}^{d-1}$), and Lemma 1498 (cardinal of $\mathcal{A}_k^d$).

**Cardinal. Case $i=0$.** Direct consequence of Definition 1486 (sets of multi-indices $\mathcal{A}_k^d$ and $\mathcal{C}_k^d$), and Lemma 1495 (cardinal of $\mathcal{C}_k^d$).

**Case $i \in [1..d]$.** Direct consequence of Lemma 1590 (number of Lagrange nodes of $\mathbb{P}_k^d$, injectivity, thus $\text{card}\{\mathbf{a}_{\alpha} \in \mathcal{H}_{i}^{\|\mathbf{v}\|^d} \mid \alpha \in \mathcal{A}_k^d\} = \text{card}\{\mathcal{A}_{k,i}^d\}$), Lemma 1501 (cardinal of $\mathcal{A}_{k,i}^d$ and $\mathcal{A}_{k,i}^{d-1}$), Lemma 1500 (cardinal of $\mathcal{C}_k^d$ and $\mathcal{A}_{k,i}^{d-1}$), and Lemma 1495 (cardinal of $\mathcal{C}_k^d$). □

*Remark 1606.* In the next lemma, we recall that $\theta_i^{d-1}$ is defined in Lemma 1368, $f_{k,i}^d$ in Lemmas 1500 ($i=0$) and 1501 ($i \in [1..d]$), and $\phi_{\pi_l}^{\|\mathbf{v}\|^d}$ in Definition 1578. See also Figure 9.6.
**Lemma 1607 (image of nodes by geometric hyperface mapping).** Let $d \ge 2$. Let $k \ge 1$. Let $\llbracket \mathbf{v} \rrbracket^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$. Let $\llbracket \hat{\mathbf{a}}^{d-1} \rrbracket_{\mathcal{A}_k^{d-1}}$ be the reference Lagrange nodes of $\mathbb{P}_k^{d-1}$ and $\llbracket \mathbf{a} \rrbracket_{\mathcal{A}_k^d}$ be the Lagrange nodes of $\mathbb{P}_k^d$ for $\llbracket \mathbf{v} \rrbracket^d$. Let $i \in [0..d]$. Let $\alpha' \in \mathcal{A}_k^{d-1}$. Then, we have

$$ (9.97) \qquad \phi_{\theta_i^{d-1}}^{\llbracket \mathbf{v} \rrbracket^d}(\hat{\mathbf{a}}_{\alpha'}^{d-1}) = \mathbf{a}_{f_{k,i}^d}(\alpha'). $$

*Proof.* From Lemma 1599 (reference Lagrange nodes of $\mathbb{P}_k^d$, (9.92) with $d-1 \ge 1$), we have

$$ \hat{\mathbf{a}}_{\alpha'}^{d-1} = \left(1 - \frac{|\alpha'|}{k}\right) \hat{\mathbf{v}}_0^{d-1} + \sum_{j=1}^{d-1} \frac{\alpha'_j}{k} \hat{\mathbf{v}}_j^{d-1} \quad \text{with} \quad \left(1 - \frac{|\alpha'|}{k}\right) + \sum_{j=1}^{d-1} \frac{\alpha'_j}{k} = 1. $$

**Case i = 0.** Then, from Lemma 1500 (cardinal of $C_k^d$ and $\mathcal{A}_k^{d-1}$), Definition 1486 (sets of multi-indices $\mathcal{A}_k^d$ and $C_k^d$), Lemma 1591 (barycentric coordinates of Lagrange nodes of $\mathbb{P}_k^d$, (9.84)), and field properties of $\mathbb{R}$ (with $k \ne 0$), we have $f_{k,0}^d(\alpha') = (k - |\alpha'|, \alpha') \in C_k^d \subset \mathcal{A}_k^d$, $|f_{k,0}^d(\alpha')| = k$,

$$ \mathbf{a}_{f_{k,0}^d}(\alpha') = \left(1 - \frac{|\alpha'|}{k}\right) \mathbf{v}_1 + \sum_{j=1}^{d-1} \frac{\alpha'_j}{k} \mathbf{v}_{j+1}. $$

**Case i ∈ [1..d].** Then, from Lemma 1501 (cardinal of $\mathcal{A}_{k,i}^d$ and $\mathcal{A}_k^{d-1}$), Definition 1486 (sets of multi-indices $\mathcal{A}_k^d$ and $C_k^d$), and Lemma 1591 (barycentric coordinates of Lagrange nodes of $\mathbb{P}_k^d$, (9.84)), we have $f_{k,i}^d(\alpha') = (\alpha'_1, \dots, \alpha'_{i-1}, 0, \alpha'_i, \dots, \alpha'_{d-1}) \in \mathcal{A}_{k,i}^d \subset \mathcal{A}_k^d$, $|f_{k,i}^d(\alpha')| = |\alpha'|$, and

$$ \mathbf{a}_{f_{k,i}^d}(\alpha') = \left(1 - \frac{|\alpha'|}{k}\right) \mathbf{v}_0 + \sum_{j=1}^{i-1} \frac{\alpha'_j}{k} \mathbf{v}_j + \sum_{j=i}^{d-1} \frac{\alpha'_j}{k} \mathbf{v}_{j+1}. $$

Therefore, in both cases, $\hat{\mathbf{a}}_{\alpha'}^{d-1}$ is the barycenter of $(\hat{\mathbf{v}}_j^{d-1})_{j \in [0..d-1]}$ and $\mathbf{a}_{f_{k,i}^d}(\alpha')$ is the barycenter of $(\mathbf{v}_j)_{j \in [0..d] \setminus \{i\}}$ with the same coefficients. Thus, from Lemma 1584 (geometric hyperface mapping, affineness and (9.74), thus $\phi_{\theta_i^{d-1}}^{\llbracket \mathbf{v} \rrbracket^d}(\hat{\mathbf{v}}_j^{d-1}) = \mathbf{v}_j$ for $j < i$ and $\phi_{\theta_i^{d-1}}^{\llbracket \mathbf{v} \rrbracket^d}(\hat{\mathbf{v}}_j^{d-1}) = \mathbf{v}_{j+1}$ for $j \ge i$), and Lemma 1400 (affine map preserves barycenter), we have the equality. $\square$

## 9.7 Lagrange linear forms for $\mathbb{P}_k^d$

**Definition 1608 (Lagrange linear forms for $\mathbb{P}_k^d$).**

Let $d \ge 1$. Let $k \in \mathbb{N}$. Let $\llbracket \mathbf{v} \rrbracket^d$ be $d+1$ points in $\mathbb{R}^d$. The Lagrange linear forms associated with the Lagrange nodes of $\mathbb{P}_k^d$ are denoted $\Sigma_{\{\mathbf{a}\}_{\mathcal{A}_k^d}} = (\sigma_{\alpha})_{\alpha \in \mathcal{A}_k^d}$, and are defined by

$$ (9.98) \qquad \forall \alpha \in \mathcal{A}_k^d, \quad \forall f : \mathbb{R}^d \to \mathbb{R}, \quad \sigma_{\alpha}(f) \stackrel{\text{def.}}{=} f(\mathbf{a}_{\alpha}). $$

**Lemma 1609 (Lagrange linear forms for $\mathbb{P}_k^d$ for $d=1$ are Lagrange linear forms for $\mathbb{P}_k^1$).** Let $k \in \mathbb{N}$. Let $\llbracket \mathbf{v} \rrbracket^1 = \llbracket v \rrbracket^1$ be two points in $\mathbb{R}^1$.

Then, $\Sigma_{\{\mathbf{a}\}_{\mathcal{A}_k^d}}$ for $d \stackrel{\text{def.}}{=} 1$ and $\Sigma_{\{\mathbf{a}\}_{\mathcal{A}_k^1}}$ from Definition 1472 coincide.

*Proof.* Direct consequence of Definition 1608 (Lagrange linear forms for $\mathbb{P}_k^d$, with $d \stackrel{\text{def.}}{=} 1$), Lemma 1589 (Lagrange nodes of $\mathbb{P}_k^d$ for $d=1$ are Lagrange nodes of $\mathbb{P}_k^1$), Lemma 1497 (first multi-indices $\mathcal{A}_k^d$, with $d \stackrel{\text{def.}}{=} 1$), and Definition 1472 (Lagrange linear forms for $\mathbb{P}_k^1$). $\square$
**Lemma 1610 (Lagrange linear forms of $\mathbb{P}_k^d$ are linear).**

Let $d \ge 1$. Let $k \in \mathbb{N}$. Let $[[v]]^d$ be $d+1$ points in $\mathbb{R}^d$. Let $\alpha \in \mathcal{A}_k^d$. Then, $\sigma_\alpha$ is linear.

*Proof.* Direct consequence of Definition 1608 (Lagrange linear forms for $\mathbb{P}_k^d$), and **the definition of linear operations over functions**. □

**Lemma 1611 (cardinal of Lagrange linear forms of $\mathbb{P}_k^d$).**

Let $d \ge 1$. Let $k \in \mathbb{N}$.

Let $[[v]]^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$. Then, $\text{card}(\Sigma^{\{a\}}_{\mathcal{A}_k^d}) = \binom{k+d}{d}$.

*Proof.* Direct consequence of Definition 1608 (Lagrange linear forms for $\mathbb{P}_k^d$), and Lemma 1590 (number of Lagrange nodes of $\mathbb{P}_k^d$). □

**Definition 1612 (reference Lagrange linear forms for $\mathbb{P}_k^d$).**

Let $d \ge 1$. Let $k \in \mathbb{N}$. The reference Lagrange linear forms associated with the reference Lagrange nodes of $\mathbb{P}_k^d$ are denoted $\hat{\Sigma}_k^d = (\hat{\sigma}_\alpha)_{\alpha \in \mathcal{A}_k^d}$, and are defined by

$$ (9.99) \qquad \forall \alpha \in \mathcal{A}_k^d, \quad \forall \hat{f} : \mathbb{R}^d \to \mathbb{R}, \quad \hat{\sigma}_\alpha(\hat{f}) \stackrel{\text{def.}}{=} \hat{f}(\hat{\alpha}_\alpha). $$

**Lemma 1613 (reference Lagrange linear forms for $\mathbb{P}_k^d$ for $d=1$ are reference Lagrange linear forms for $\mathbb{P}_k^1$).** Let $k \in \mathbb{N}$. Then, $\hat{\Sigma}_k^d$ for $d \stackrel{\text{def.}}{=} 1$ and $\hat{\Sigma}_k^1$ from Definition 1456 coincide.

*Proof.* Direct consequence of Definition 1612 (reference Lagrange linear forms for $\mathbb{P}_k^d$, with $d \stackrel{\text{def.}}{=} 1$), Lemma 1600 (reference Lagrange nodes of $\mathbb{P}_k^d$ for $d=1$ are reference Lagrange nodes of $\mathbb{P}_k^1$), Lemma 1497 (first multi-indices $\mathcal{A}_k^d$, with $d \stackrel{\text{def.}}{=} 1$), and Definition 1456 (reference Lagrange linear forms for $\mathbb{P}_k^1$). □

**Lemma 1614 (Lagrange linear forms of $\mathbb{P}_k^d$ are images of reference).**

Let $d \ge 1$. Let $k \in \mathbb{N}$. Let $[[v]]^d$ be $d+1$ points in $\mathbb{R}^d$. Let $\alpha \in \mathcal{A}_k^d$. Let $f : \mathbb{R}^d \to \mathbb{R}$.

Then, we have $\sigma_\alpha(f) = \hat{\sigma}_\alpha(\hat{f})$ where $\hat{f} = f \circ \varphi_{\text{geo}}^{[[v]]^d}$.

*Proof.* Direct consequence of Definition 1608 (Lagrange linear forms for $\mathbb{P}_k^d$), Lemma 1604 (Lagrange nodes of $\mathbb{P}_k^d$ are image of reference), and Definition 1612 (reference Lagrange linear forms for $\mathbb{P}_k^d$). □

## 9.8 Unisolvence for $\text{Lag}\mathbb{P}_0^d$

**Lemma 1615 (Lagrange linear forms for $\mathbb{P}_0^d$ are injective).**

Let $d \ge 1$. Let $[[v]]^d$ be $d+1$ points of $\mathbb{R}^d$. Then, $\phi_{\Sigma^{\{a\}}_{\mathcal{A}_0^d}}$ is injective.

*Proof.* Direct consequence of Definition 1424 (finite element triple), Lemma 1509 (constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$, thus for all $p \in \mathbb{P}_0^d$, there exists $a_0 \in \mathbb{R}$ such that $p = (x \mapsto a_0)$), Definition 1608 (Lagrange linear forms for $\mathbb{P}_k^d$, thus $\sigma_\alpha(p) = p(\mathbf{a}_\alpha) = a_0 = 0$, and $p = 0$), Definition 101 (kernel), and Lemma 103 (injective linear map has zero kernel). □

*Remark 1616.* In the next Lemma, the hypothesis of affinely independent points is not necessary. In fact, Lemmas 1611 and 1590 do not require this hypothesis when $k \stackrel{\text{def.}}{=} 0$, as there is a single node set at the isobarycenter of $[[v]]^d$, which is always defined.

However, to have a proper FE, it is required to have a nondegenerate simplex, and thus the hypothesis remains necessary at that stage. This is why we did not relax it here (otherwise, the statements in Lemmas 1611 and 1590 would have been more intricate).
**Lemma 1617 (unisolvence of $\mathbb{P}_0^d$).**

Let $d \ge 1$. Let $[[v]]^d$ be $d+1$ affinely independent points of $\mathbb{R}^d$.

Then, $(K^{[[v]]^d}, \mathbb{P}_0^d, \Sigma^{\{a\}}_{\mathcal{A}_0^d})$ satisfies the unisolvence property.

*Proof.* Direct consequence of Lemma 1525 (*dimension of $\mathbb{P}_k^d$*), Lemma 1611 (*cardinal of Lagrange linear forms of $\mathbb{P}_k^d$, with $k \stackrel{\text{def.}}{=} 0$*), Lemma 1364 (*properties of the binomial coefficient*, (5.4)), **the definition of order (reflexivity, thus $\dim \mathbb{P}_0^d = 1 \ge 1 = \text{card} (\Sigma^{\{a\}}_{\mathcal{A}_0^d})$**), Lemma 1615 (*Lagrange linear forms for $\mathbb{P}_0^d$ are injective*), and Lemma 1426 (*injectivity implies unisolvence*). $\square$

## 9.9 Unisolvence for $\text{Lag}\mathbb{P}_1^d$

**Lemma 1618 (decomposition of $\mathbb{P}_1^d$ polynomial with $\sigma_\alpha$).**

Let $d \ge 1$. Let $[[v]]^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$. Let $p \in \mathbb{P}_1^d$. Then, we have

$$ (9.100) \qquad p = \sigma_0(p) \lambda_0^{[[v]]^d} + \sum_{i=1}^d \sigma_{e_i}(p) \lambda_i^{[[v]]^d} = \sigma_0(p) \mathcal{L}_0^{[[v]]^d} + \sum_{i=1}^d \sigma_{e_i}(p) \mathcal{L}_i^{[[v]]^d}. $$

*Proof.* Direct consequence of Definition 1608 (*Lagrange linear forms for $\mathbb{P}_k^d$*), Lemma 1593 (*Lagrange nodes of $\mathbb{P}_1^d$ are vertices*, thus $\sigma_0(p) = p(v_0)$, and $\sigma_{e_i}(p) = p(v_i)$ for all $i \in [1..d]$), Lemma 1561 (*decomposition of $\mathbb{P}_1^d$ polynomial with barycentric coordinates*), and Lemma 1555 (*composition of $\mathbb{P}_1^d$ polynomial in Lagrange basis*). $\square$

**Lemma 1619 (Lagrange linear forms for $\mathbb{P}_1^d$ are injective).**

Let $d \ge 1$. Let $[[v]]^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$. Then, $\phi_{\Sigma^{\{a\}}_{\mathcal{A}_1^d}}$ is injective.

*Proof.* Direct consequence of Definition 1424 (*finite element triple*), Lemma 1497 (*first multi-indices $\mathcal{A}_k^d$, with $k \stackrel{\text{def.}}{=} 1$*), Lemma 1618 (*decomposition of $\mathbb{P}_1^d$ polynomial with $\sigma_\alpha$, thus $\forall \alpha \in \mathcal{A}_1^d$, $\sigma_\alpha(p) = 0$ implies $p = 0$*), Definition 101 (*kernel*), and Lemma 103 (*injective linear map has zero kernel*). $\square$

**Lemma 1620 (unisolvence of $\mathbb{P}_1^d$).** Let $d \ge 1$. Let $[[v]]^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$. Then, $(K^{[[v]]^d}, \mathbb{P}_1^d, \Sigma^{\{a\}}_{\mathcal{A}_1^d})$ satisfies the unisolvence property.

*Proof.* Direct consequence of Lemma 1525 (*dimension of $\mathbb{P}_k^d$*), Lemma 1611 (*cardinal of Lagrange linear forms of $\mathbb{P}_k^d$, with $k \stackrel{\text{def.}}{=} 1$*), Lemma 1364 (*properties of the binomial coefficient*, (5.5) with $n \stackrel{\text{def.}}{=} d+1$), **the definition of order (reflexivity, thus $\dim \mathbb{P}_1^d = d+1$ is greater than or equal to $d+1 = \text{card} (\Sigma^{\{a\}}_{\mathcal{A}_1^d})$**), Lemma 1619 (*Lagrange linear forms for $\mathbb{P}_1^d$ are injective*), and Lemma 1426 (*injectivity implies unisolvence*). $\square$

## 9.10 Unisolvence for $\text{Lag}\mathbb{P}_k^d$

**Lemma 1621 (factorization of zero polynomial on last reference hyperplane).**

Let $d \ge 1$. Let $k \ge 1$. Let $\hat{p} \in \mathbb{P}_k^d$. Then, we have the equivalence

$$ (9.101) \qquad \hat{p}|_{\hat{\mathcal{H}}_d^d} = 0 \iff \exists \hat{q} \in \mathbb{P}_{k-1}^d, \quad \hat{p} = \hat{\mathcal{L}}_d^{1,d} \hat{q}. $$
**Proof.** **From right to left.** Direct consequence of Lemma 1564 (reference face hyperplane, (9.59) with $i \stackrel{\text{def.}}{=} d$), Definition 101 (kernel), and **the zero-product property in $\mathbb{R}$**.

**From left to right.** From Lemma 1529 (decomposition of $\mathbb{P}_k^d$), Lemma 1564 (reference face hyperplane, (9.59) with $i \stackrel{\text{def.}}{=} d$), and Definition 1541 (reference Lagrange polynomials of $\mathbb{P}_1^d$, (9.30) with $i \stackrel{\text{def.}}{=} d$), there exists $\tilde{p}_0 \in \mathbb{P}_k^{d-1}$ and $\hat{q} \in \mathbb{P}_{k-1}^d$ such that $\hat{p} = \tilde{p}_0 + X_d \hat{q}$, thus for all $\hat{x} = (\hat{x}_1, \dots, \hat{x}_{d-1}, 0) \in \hat{\mathcal{H}}_d^d$, we have

$$0 = \hat{p}(\hat{x}) = \tilde{p}_0(\hat{x}_1, \dots, \hat{x}_{d-1}) + 0 \hat{q}(\hat{x}_1, \dots, \hat{x}_{d-1}, 0) = \tilde{p}_0(\hat{x}_1, \dots, \hat{x}_{d-1}),$$

i.e. $\tilde{p}_0 = 0$, and $\hat{p} = \hat{\mathcal{L}}_d^{1,d} \hat{q}$.

**Remark 1622.** In the second part of the proof of the next lemma, we used the circular permutation $c_i^d$, see Lemma 1366. The geometric mapping with permutation $\phi_{c_i^d}^{[\mathbf{v}]^d}$ is illustrated in Figure 9.7 when $i=0$ and $d=3$.

Note that one could also use for $\pi^d$ the transposition $\tau_i^d$ exchanging $i$ and $d$, see Lemma 1367, which is also injective and satisfies $\tau_i^d(d) = i$.

**Lemma 1623 (factorization of zero polynomial on hyperplane $\mathbb{P}_k^d$).** Let $d \ge 1$. Let $k \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$. Let $p \in \mathbb{P}_k^d$. Let $i \in [0..d]$. Then, we have the equivalence

$$(9.102) \qquad p_{|\mathcal{H}_i^{[\mathbf{v}]^d}} = 0 \iff \exists q \in \mathbb{P}_{k-1}^d, \quad p = \lambda_i^{[\mathbf{v}]^d} q.$$

**Proof.** **From right to left.** Direct consequence of Lemma 1563 (equivalent definition of face hyperplane, (9.57) with $i \stackrel{\text{def.}}{=} 0$), Definition 101 (kernel), and **the zero-product property in $\mathbb{R}$**.

**From left to right.** Assume that $p_{|\mathcal{H}_i^{[\mathbf{v}]^d}} = 0$. Let $\hat{p} \stackrel{\text{def.}}{=} p \circ \phi_{c_i^d}^{[\mathbf{v}]^d}$.

Then, from Lemma 1366 (circular permutation, thus $c_i^d$ bijective, $c_i^d(d) = i$ and $(c_i^d)^{-1}(i) = d$), **the definition of bijectivity (implies injectivity)**, Lemma 1586 (geometric mapping with permutation, with $\pi^d \stackrel{\text{def.}}{=} c_i^d$, (9.78) with $j \stackrel{\text{def.}}{=} d$, thus $\phi_{c_i^d}^{[\mathbf{v}]^d}(\hat{\mathcal{H}}_d^d) = \mathcal{H}_i^{[\mathbf{v}]^d}$ and $\hat{p}_{|\hat{\mathcal{H}}_d^d} = 0$), Lemma 1621 (factorization of zero polynomial on last reference hyperplane), Lemma 1586 (geometric mapping with permutation, thus $\phi_{c_i^d}^{[\mathbf{v}]^d}$ is affine and bijective, and (9.77) with $i \stackrel{\text{def.}}{=} i$), **the rules of composition with a bijective function**, Lemma 1410 (inverse of affine map is affine map), and Lemma 1540 (affine mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$, with $l \stackrel{\text{def.}}{=} d$, $k \stackrel{\text{def.}}{=} k-1$ and $f \stackrel{\text{def.}}{=} (\phi_{c_i^d}^{[\mathbf{v}]^d})^{-1}$ affine), there exists $\hat{q} \in \mathbb{P}_{k-1}^d$ such that $\hat{p} = \hat{\mathcal{L}}_d^{1,d} \hat{q}$, and thus we have

$$p = \left( \hat{\mathcal{L}}_{(c_i^d)^{-1}(i)}^{1,d} \circ (\phi_{c_i^d}^{[\mathbf{v}]^d})^{-1} \right) \left( \hat{q} \circ (\phi_{c_i^d}^{[\mathbf{v}]^d})^{-1} \right) = \lambda_i^{[\mathbf{v}]^d} q,$$

with $q \stackrel{\text{def.}}{=} \hat{q} \circ (\phi_{c_i^d}^{[\mathbf{v}]^d})^{-1} \in \mathbb{P}_{k-1}^d$.

**Remark 1624.** See the sketch of the next two proofs in Section 3.1.

The first step dealing with the induction from dimension $d-1$ to $d$ (with constant degree $k$) is illustrated in Figure 9.6. The second step that passes from degree $k-1$ to $k$ (with constant dimension $d$) is illustrated in Figure 9.8.

**Lemma 1625 (Lagrange linear forms for $\mathbb{P}_k^d$ are injective).** Let $d \ge 1$. Let $k \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$. Then, $\phi_{\sum \{a\}_{A_k^d}}$ is injective.
**Proof.** For all $d \ge 1$ and $k \ge 1$, let $P$ be the predicate defined by

$$ P(d, k) \stackrel{\text{def.}}{=} \left[ \forall \llbracket \mathbf{v} \rrbracket^d \in \mathbb{R}^d, \quad \llbracket \mathbf{v} \rrbracket^d \text{ affinely independent} \quad \Rightarrow \quad \phi_{\sum \{\mathbf{a}\}_{A_k^d}}^{\{\mathbf{a}\}_{A_k^d}} \text{ is injective} \right] $$

where the application $\phi_{\sum \{\mathbf{a}\}_{A_k^d}}$ is defined in Definition 1424 (finite element triple), the Lagrange linear forms $\sum_{\{\mathbf{a}\}_{A_k^d}}$ are defined in Definition 1608 (Lagrange linear forms for $\mathbb{P}_k^d$), and the Lagrange nodes $\{\mathbf{a}\}_{A_k^d}$ are defined in Definition 1588 (Lagrange nodes of $\mathbb{P}_k^d$).

**Double induction: $P(1, k)$, for all $k \ge 1$.**

Direct consequence of Lemma 1413 (affinely independent family of 2 elements, thus $v_0 \ne v_1$), Lemma 1476 (Lagrange linear forms for $\mathbb{P}_k^1$ are injective), and Lemma 1609 (Lagrange linear forms for $\mathbb{P}_k^d$ for $d=1$ are Lagrange linear forms for $\mathbb{P}_k^1$).

**Double induction: $P(d, 1)$, for all $d \ge 1$.**

Direct consequence of Lemma 1619 (Lagrange linear forms for $\mathbb{P}_1^d$ are injective).

**Double induction: $P(d-1, k) \wedge P(d, k-1) \Rightarrow P(d, k)$, for all $d, k \ge 2$.**

Let $d \ge 2$. Let $k \ge 2$. Assume that $P(d-1, k)$ and $P(d, k-1)$ hold.

Let $\llbracket \mathbf{v} \rrbracket^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$. Let $p \in \mathbb{P}_k^d$ such that for all $\alpha \in \mathcal{A}_k^d$, $p(\mathbf{a}_\alpha) = 0$ where the Lagrange nodes $\{\mathbf{a}\}_{\mathcal{A}_k^d}$ are defined in Definition 1588 (Lagrange nodes of $\mathbb{P}_k^d$).

**Step 1: factorization using $P(d-1, k)$.** Let $p_0 \stackrel{\text{def.}}{=} p \circ \phi_{\theta_0^{d-1}}^{\llbracket \mathbf{v} \rrbracket^d}$.

Let $\alpha' \in \mathcal{A}_k^{d-1}$. Then, from Lemma 1585 (hyperface geometric mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^{d-1}$, with $i \stackrel{\text{def.}}{=} 0$, thus $p_0 \in \mathbb{P}_k^{d-1}$), Lemma 1496 ($C_k^d$ are layers of $\mathcal{A}_k^d$), Lemma 1500 (cardinal of $C_k^d$ and $\mathcal{A}_k^{d-1}$, thus $f_{k,0}^d(\alpha') \in C_k^d \subset \mathcal{A}_k^d$), Lemma 1607 (image of nodes by geometric hyperface mapping, with $i \stackrel{\text{def.}}{=} 0$), and the assumption on $p$, we have $p_0(\hat{\mathbf{a}}_{\alpha'}^{d-1}) = p(\mathbf{a}_{f_{k,0}(\alpha')}) = 0$.

Then, from Lemma 1436 (family of reference points is affinely independent, with $d-1 \ge 1$), and the property $P(d-1, k)$ applied to the reference vertices $\llbracket \hat{\mathbf{v}}^{d-1} \rrbracket^{d-1}$ (thus using the reference nodes $\{\hat{\mathbf{a}}\}_{\mathcal{A}_k^{d-1}}$, $\phi_{\sum \{\mathbf{a}\}_{\mathcal{A}_k^{d-1}}}^{\{\mathbf{a}\}_{\mathcal{A}_k^{d-1}}}$ is injective. Hence, from Lemma 103 (injective linear map has zero kernel), Definition 101 (kernel), Definition 1424 (finite element triple), and Definition 1608 (Lagrange linear forms for $\mathbb{P}_k^d$), we have $\phi_{\sum \{\mathbf{a}\}_{\mathcal{A}_k^{d-1}}} (p_0) = (p_0(\hat{\mathbf{a}}_{\alpha'}^{d-1}))_{\alpha' \in \mathcal{A}_k^{d-1}} = 0$ implies $p_0 = 0$.

Finally, from Lemma 1584 (geometric hyperface mapping, with $i \stackrel{\text{def.}}{=} 0$, bijection from $\mathbb{R}^{d-1}$ onto $\mathcal{H}_0^{\llbracket \mathbf{v} \rrbracket^d}$), the rules of composition with a bijective function, and Lemma 1623 (factorization of zero polynomial on hyperplane $\mathbb{P}_k^d$, with $i \stackrel{\text{def.}}{=} 0$), we have

$$ \forall \mathbf{x} \in \mathcal{H}_0^{\llbracket \mathbf{v} \rrbracket^d}, \quad p(\mathbf{x}) = p_0 \circ (\phi_{\theta_0^{d-1}}^{\llbracket \mathbf{v} \rrbracket^d})^{-1}(\mathbf{x}) = 0, $$

i.e. $p_{|\mathcal{H}_0^{\llbracket \mathbf{v} \rrbracket^d}} = 0$, and there exists $q \in \mathbb{P}_{k-1}^d$ such that $p = \lambda_0^{\llbracket \mathbf{v} \rrbracket^d} q$.

**Step 2: cancellation using $P(d, k-1)$.** Let $\alpha \in \mathcal{A}_{k-1}^d$.

Then, from Lemma 1496 ($C_k^d$ are layers of $\mathcal{A}_k^d$, thus $\alpha \in \mathcal{A}_k^d$ and $\mathcal{A}_k^d \setminus C_k^d = \mathcal{A}_{k-1}^d$), Lemma 1605 (face hyperplanes of Lagrange nodes of $\mathbb{P}_k^d$, (9.95), contrapositive, thus $\mathbf{a}_\alpha \notin \mathcal{H}_0^{\llbracket \mathbf{v} \rrbracket^d}$), Definition 101 (kernel), Lemma 1563 (equivalent definition of face hyperplane, (9.57) with $i \stackrel{\text{def.}}{=} 0$, contrapositive, thus $\lambda_0^{\llbracket \mathbf{v} \rrbracket^d}(\mathbf{a}_\alpha) \ne 0$), the assumption on $p$, and the zero-product property in $\mathbb{R}$ (applied to $\lambda_0^{\llbracket \mathbf{v} \rrbracket^d}(\mathbf{a}_\alpha) q(\mathbf{a}_\alpha) = 0$), we have $q(\mathbf{a}_\alpha) = 0$. Thus, from Lemma 1598 ($\mathbb{P}_{k-1}^d$ sub-nodes of sub-vertices are some nodes of $\mathbb{P}_{k-1}^d$), we have $q(\mathbf{a}_\alpha) = 0$.

Finally, from Lemma 1597 (sub-vertices are affinely independent), and the property $P(d, k-1)$ applied to the sub-vertices $\llbracket \mathbf{v} \rrbracket^d$ (thus using the $\mathbb{P}_{k-1}^d$ sub-nodes $\{\mathbf{a}\}_{\mathcal{A}_{k-1}^d}$), $\phi_{\sum \{\mathbf{a}\}_{\mathcal{A}_{k-1}^d}}^{\{\mathbf{a}\}_{\mathcal{A}_{k-1}^d}}$ is injective. Hence, from Lemma 103 (injective linear map has zero kernel), Definition 101 (kernel), and
Definition 1424 (finite element triple), we have

$$ \phi_{\sum_{\mathcal{A}_{k-1}^{d}} \{\mathbf{a}\}}(q) = (q(\mathbf{a}_{\alpha}))_{\alpha \in \mathcal{A}_{k-1}^{d}} = 0 \implies q = 0. $$

Thus, we also have $p = 0$, and from Lemma 103 (injective linear map has zero kernel), Definition 101 (kernel), and Definition 1424 (finite element triple), $\phi_{\sum_{\mathcal{A}_{k}^{d}} \{\mathbf{a}\}}$ is injective, i.e. the property $P(d, k)$ holds, which concludes the double induction.

Therefore, from Lemma 1358 (double induction by diagonal, starting from 1 for both $d$ and $k$), the property $P(d, k)$ holds for all $d, k \ge 1$. $\square$

**Theorem 1626 (unisolvence of $\mathbb{P}_k^d$).** Let $d \ge 1$. Let $k \ge 1$. Let $[\mathbf{v}]^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$. Then, $(K^{[\mathbf{v}]^d}, \mathbb{P}_k^d, \sum_{\mathcal{A}_k^d} \{\mathbf{a}\}})$ satisfies the unisolvence property.

*Proof.* Direct consequence of Lemma 1525 (dimension of $\mathbb{P}_k^d$), Lemma 1611 (cardinal of Lagrange linear forms of $\mathbb{P}_k^d$), the definition of order (reflexivity, thus $\dim \mathbb{P}_k^d \ge \text{card} (\sum_{\mathcal{A}_k^d} \{\mathbf{a}\})$), Definition 1608 (Lagrange linear forms for $\mathbb{P}_k^d$), Lemma 1625 (Lagrange linear forms for $\mathbb{P}_k^d$ are injective), and Lemma 1426 (injectivity implies unisolvence). $\square$

**Remark 1627.**

The proof of next lemma (from left to right) is similar to step 1 in the proof of Lemma 1625.

This face unisolvence result allows to ensure the continuity of functions that are piecewise $\mathbb{P}_k^d$ in each element (assuming that the mesh is conforming). Indeed, let $f$ be a function that is in $\mathbb{P}_k^d$ in two adjacent mesh cells $K$ and $K'$, sharing a face $\mathcal{F} = K \cap K'$. To have $f$ continuous, it suffices to enforce that the two polynomials $f|_K$ and $f|_{K'}$ take the same values at the nodes of the face $\mathcal{F}$.

**Lemma 1628 (face unisolvence of $\mathbb{P}_k^d$).** Let $d \ge 2$. Let $k \ge 1$. Let $\mathcal{A}_{k,0}^d \stackrel{\text{def.}}{=} \mathcal{C}_k^d$.

Let $[\mathbf{v}]^d$ be $d+1$ affinely independent points in $\mathbb{R}^d$. Let $p \in \mathbb{P}_k^d$. Let $i \in [0..d]$.

Then, we have the face unisolvence property:

$$ (9.103) \qquad (\forall \alpha \in \mathcal{A}_{k,i}^d, \quad p(\mathbf{a}_\alpha) = 0) \iff p_{|\mathcal{H}_i^{[\mathbf{v}]^d}} = 0. $$

*Proof. From right to left.*

Direct consequence of Lemma 1605 (face hyperplanes of Lagrange nodes of $\mathbb{P}_k^d$, thus $\mathbf{a}_\alpha \in \mathcal{H}_i^{[\mathbf{v}]^d}$).

**From left to right.** Assume that for all $\alpha \in \mathcal{A}_{k,i}^d$, we have $p(\mathbf{a}_\alpha) = 0$. Let $p_i \stackrel{\text{def.}}{=} p \circ \phi_{\theta^{d-1}}^{[\mathbf{v}]^d}$.

Let $\alpha' \in \mathcal{A}_{k,i}^{d-1}$. Then, from Lemma 1585 (hyperface geometric mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^{d-1}$, thus $p_i \in \mathbb{P}_k^{d-1}$), Lemma 1500 (cardinal of $\mathcal{C}_k^d$ and $\mathcal{A}_k^{d-1}$, for $i = 0$, thus $f_{k,0}^d(\alpha') \in \mathcal{A}_{k,0}^d$), Lemma 1501 (cardinal of $\mathcal{A}_{k,i}^d$ and $\mathcal{A}_k^{d-1}$, for $i \in [1..d]$, thus $f_{k,i}^d(\alpha') \in \mathcal{A}_{k,i}^d$), Lemma 1607 (image of nodes by geometric hyperface mapping), and the assumption on $p$, we have $p_i(\hat{\mathbf{a}}_{\alpha'}^{d-1}) = p(\mathbf{a}_{f_{k,i}^d(\alpha')}^{d-1}) = 0$.

Then, from Lemma 1436 (family of reference points is affinely independent, with $d-1 \ge 1$), Lemma 1625 (Lagrange linear forms for $\mathbb{P}_k^d$ are injective, with $d-1 \ge 1$ and $[\hat{\mathbf{v}}^{d-1}]^{d-1}$), Lemma 103 (injective linear map has zero kernel), Definition 101 (kernel), Definition 1424 (finite element triple), and Definition 1608 (Lagrange linear forms for $\mathbb{P}_k^d$), $\phi_{\sum_{\mathcal{A}_k^{d-1}} \{\mathbf{a}\}}$ is injective, and thus $\phi_{\sum_{\mathcal{A}_k^{d-1}} \{\mathbf{a}\}}(p_i) = (p_i(\hat{\mathbf{a}}_{\alpha'}^{d-1}))_{\alpha' \in \mathcal{A}_k^{d-1}} = 0$ implies $p_i = 0$.

Finally, from Lemma 1584 (geometric hyperface mapping, bijection from $\mathbb{R}^{d-1}$ onto $\mathcal{H}_i^{[\mathbf{v}]^d}$), and the rules of composition with a bijective function, we have

$$ \forall \mathbf{x} \in \mathcal{H}_i^{[\mathbf{v}]^d}, \quad p(\mathbf{x}) = p_i \circ (\phi_{\theta^{d-1}}^{[\mathbf{v}]^d})^{-1}(\mathbf{x}) = 0, $$

$$ \text{i.e. } p_{|\mathcal{H}_i^{[\mathbf{v}]^d}} = 0. \qquad \square $$
## 9.11 $\text{Log}\mathbb{P}_k^d$ Lagrange finite element

**Theorem 1629** ($\text{Log}\mathbb{P}_k^d$ Lagrange finite element). Let $d \ge 1$. Let $k \in \mathbb{N}$. Let $[[\mathbf{v}]]^d$ be $d+1$ affinely independent points of $\mathbb{R}^d$. Then, $\text{Log}\mathbb{P}_k^d \stackrel{\text{def.}}{=} (K[[\mathbf{v}]^d], \mathbb{P}_k^d, \Sigma^{\{\mathbf{a}\}}_{\mathcal{A}_k^d})$ is a finite element.

It is called the Lagrange finite element of degree $k$ associated with vertices $[[\mathbf{v}]^d]$.

*Proof.* Direct consequence of Lemma 1557 (nontrivial simplex), Lemma 1525 (dimension of $\mathbb{P}_k^d$, thus $\dim \mathbb{P}_k^d \ge 1$ and $\mathbb{P}_k^d \ne \{0\}$), Lemma 1617 (unisolvence of $\mathbb{P}_0^d$, for $k=0$), Theorem 1626 (unisolvence of $\mathbb{P}_k^d$, for $k \ge 1$), and Definition 1424 (finite element triple, with $q \stackrel{\text{def.}}{=} 1$). $\square$

**Lemma 1630** ($\text{Log}\mathbb{P}_k^d$ Lagrange finite element for $d=1$ is Lagrange finite element on a current segment). Let $k \in \mathbb{N}$. Let $[[\mathbf{v}]^1 = [[v]^1]$ be two points in $\mathbb{R}^1$.
Then, $\text{Log}\mathbb{P}_k^d$ for $d \stackrel{\text{def.}}{=} 1$ and $\text{Log}\mathbb{P}_k^1$ from Theorem 1478 coincide.

*Proof.* Direct consequence of Theorem 1629 ($\text{Log}\mathbb{P}_k^d$ Lagrange finite element, with $d \stackrel{\text{def.}}{=} 1$), Lemma 1609 (Lagrange linear forms for $\mathbb{P}_k^d$ for $d=1$ are Lagrange linear forms for $\mathbb{P}_k^1$), and Theorem 1478 ($\text{Log}\mathbb{P}_k^1$ Lagrange finite element). $\square$

**Theorem 1631** ($\text{Log}\hat{\mathbb{P}}_k^d$ reference Lagrange finite element). Let $d \ge 1$. Let $k \in \mathbb{N}$. Then, $\text{Log}\hat{\mathbb{P}}_k^d \stackrel{\text{def.}}{=} (\hat{K}^d, \mathbb{P}_k^d, \hat{\Sigma}_k^d)$ is a finite element.

It is called the reference Lagrange finite element of degree $k$ in dimension $d$.

*Proof.* Direct consequence of Lemma 1436 (family of reference points is affinely independent), Lemma 1443 (simplex of reference vertices is reference simplex), Lemma 1599 (reference Lagrange nodes of $\mathbb{P}_k^d$), Definition 1612 (reference Lagrange linear forms for $\mathbb{P}_k^d$, thus $\hat{\Sigma}_k^d = \Sigma^{\{\hat{\mathbf{a}}\}}_{\mathcal{A}_k^d}$), and Theorem 1629 ($\text{Log}\mathbb{P}_k^d$ Lagrange finite element). $\square$

**Lemma 1632** ($\text{Log}\hat{\mathbb{P}}_k^d$ reference Lagrange finite element for $d=1$ is Lagrange finite element on the reference segment). Let $k \in \mathbb{N}$. Then, $\text{Log}\hat{\mathbb{P}}_k^d$ for $d \stackrel{\text{def.}}{=} 1$ and $\text{Log}\hat{\mathbb{P}}_k^1$ from Theorem 1460 coincide.

*Proof.* Direct consequence of Theorem 1631 ($\text{Log}\hat{\mathbb{P}}_k^d$ reference Lagrange finite element, with $d \stackrel{\text{def.}}{=} 1$), Lemma 1613 (reference Lagrange linear forms for $\mathbb{P}_k^d$ for $d=1$ are reference Lagrange linear forms for $\mathbb{P}_k^1$), and Theorem 1460 ($\text{Log}\hat{\mathbb{P}}_k^1$ reference Lagrange finite element). $\square$
# Chapter 10

# Conclusions, perspectives

We have presented very detailed proofs for the building of the simplicial Lagrange finite element (FE) in any nonzero dimension *d*, and of any order *k*. This includes the general definition of FE, some results about simplicial geometry, the simplified construction in the monodimensional case on a segment (mainly for didactic purposes, but also to prepare the tensorization in *d*-cuboids), the construction of multi-indices of given maximum length, results about multi-variate polynomials, $\mathbb{F}_1^d$ Lagrange polynomials, affine geometric mappings, Lagrange nodes and Lagrange linear forms of $\text{Log} \mathbb{P}_k^d$, and finally the proofs of unisolvence of $\text{Log} \mathbb{P}_k^d$, and of face unisolvence.

The short-term purpose of this work was to help the formalization in a formal proof assistant such as Coq of the basic concepts of FE, and the building of the simplest example: $\text{Log} \mathbb{P}_k^d$, that is emblematic and widely used in practice. First milestones towards this will be [30, 9] where special attention will be paid to the formalization of multi-indices.

Our mid-term purpose is now to continue up to the formalization of the quadrangular and hexahedric Lagrange FE, then of other (non-nodal) families of FE such as the face-flux driven Raviart-Thomas FE.

The long-term purpose of these studies is the formal proof of programs implementing the finite element method. As a consequence, after having addressed the formalization of the Lax-Milgram theorem [14, 6], and the formalization of parts of measure theory and Lebesgue integration of nonnegative functions [15, 7, 8], we will also have to write very detailed pen-and-paper proofs for $L^p$ Lebesgue spaces and $W^{m,p}$ Sobolev spaces as Banach spaces, including parts of the distribution theory, and the concepts and results of the interpolation and approximation theory to define the Finite Element Method itself.
# Acknowledgment

The authors thank Alexandre Ern for fruitful discussions, especially about unisolvence issues.
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List of Figures

<table><tr><td>1.1</td><td>Components of a typical (stationary) FEM formulation.</td><td>10</td></tr><tr><td>3.1</td><td>Lagrange nodes of the reference simplex</td><td>25</td></tr><tr><td>3.2</td><td>Lex, colex, symlex, and revlex orderings</td><td>26</td></tr><tr><td>3.3</td><td>Grlex ordering</td><td>27</td></tr><tr><td>3.4</td><td>Grcolex ordering</td><td>28</td></tr><tr><td>3.5</td><td>Grsymlex ordering</td><td>29</td></tr><tr><td>3.6</td><td>Grevlex ordering</td><td>30</td></tr><tr><td>5.1</td><td>Double induction scheme "by diagonal"</td><td>38</td></tr><tr><td>5.2</td><td>Strong double induction scheme</td><td>38</td></tr><tr><td>9.1</td><td>Lagrange nodes of the reference simplex</td><td>66</td></tr><tr><td>9.2</td><td>Vertical and horizontal slices</td><td>68</td></tr><tr><td>9.3</td><td>Mappings f<sup>d</sup><sub>k,0</sub> and f<sup>d</sup><sub>k,1</sub></td><td>71</td></tr><tr><td>9.4</td><td>Geometric mapping.</td><td>83</td></tr><tr><td>9.5</td><td>Geometric transformation</td><td>88</td></tr><tr><td>9.6</td><td>Geometric hyperface mapping</td><td>95</td></tr><tr><td>9.7</td><td>Geometric transformation with a circular permutation</td><td>96</td></tr><tr><td>9.8</td><td>Sub-vertices and Lagrange sub-nodes</td><td>99</td></tr></table>
# Appendix A

## Lists of statements

This appendix collects the references (name and number) for all statements present in Part II. These are split into definitions, lemmas, and theorems.

### List of Definitions

<table><tr><td>1362 Definition (binomial coefficient)</td><td>39</td></tr><tr><td>1365 Definition (canonic families)</td><td>40</td></tr><tr><td>1379 Definition (affine subspace)</td><td>43</td></tr><tr><td>1388 Definition (affine map)</td><td>45</td></tr><tr><td>1395 Definition (set of affine maps)</td><td>46</td></tr><tr><td>1401 Definition (isobarycenter)</td><td>47</td></tr><tr><td>1411 Definition (affinely independent family)</td><td>50</td></tr><tr><td>1416 Definition (monomial of a single variable)</td><td>51</td></tr><tr><td>1417 Definition (polynomial space P<sup>1</sup><sub>k</sub>)</td><td>51</td></tr><tr><td>1424 Definition (finite element triple)</td><td>53</td></tr><tr><td>1430 Definition (shape function)</td><td>54</td></tr><tr><td>1432 Definition (family of points)</td><td>55</td></tr><tr><td>1434 Definition (family of reference points)</td><td>55</td></tr><tr><td>1437 Definition (reference simplex)</td><td>55</td></tr><tr><td>1440 Definition (simplex)</td><td>56</td></tr><tr><td>1445 Definition (multi-indices A<sup>1</sup><sub>k</sub>)</td><td>57</td></tr><tr><td>1448 Definition (Lagrange polynomials of P<sup>1</sup><sub>k</sub>)</td><td>58</td></tr><tr><td>1453 Definition (reference Lagrange nodes of P<sup>1</sup><sub>k</sub>)</td><td>59</td></tr><tr><td>1456 Definition (reference Lagrange linear forms for P<sup>1</sup><sub>k</sub>)</td><td>59</td></tr><tr><td>1461 Definition (geometric mapping in dimension 1)</td><td>60</td></tr><tr><td>1465 Definition (Lagrange nodes of P<sup>1</sup><sub>k</sub>)</td><td>61</td></tr><tr><td>1472 Definition (Lagrange linear forms for P<sup>1</sup><sub>k</sub>)</td><td>62</td></tr><tr><td>1480 Definition (length of multi-indices)</td><td>66</td></tr><tr><td>1482 Definition (factorial of multi-indices)</td><td>66</td></tr><tr><td>1484 Definition (Kronecker delta of multi-indices)</td><td>67</td></tr><tr><td>1486 Definition (sets of multi-indices A<sup>d</sup><sub>k</sub> and C<sup>d</sup><sub>k</sub>)</td><td>67</td></tr><tr><td>1491 Definition (slices S&#771;<sup>d</sup><sub>k,i</sub> and S&#771;&#771;<sup>d</sup><sub>k,i</sub>)</td><td>68</td></tr><tr><td>1503 Definition (monomial in d variables)</td><td>72</td></tr><tr><td>1505 Definition (polynomial space P<sup>d</sup><sub>k</sub>)</td><td>72</td></tr></table>
<table><tr><td>1510 Definition (degree of polynomial)</td><td>73</td></tr><tr><td>1541 Definition (reference Lagrange polynomials of P<sup>d</sup><sub>1</sub>)</td><td>81</td></tr><tr><td>1547 Definition (geometric mapping)</td><td>82</td></tr><tr><td>1562 Definition (face hyperplane)</td><td>88</td></tr><tr><td>1566 Definition (hyperface)</td><td>90</td></tr><tr><td>1568 Definition (l-face affine space)</td><td>91</td></tr><tr><td>1572 Definition (l-face)</td><td>91</td></tr><tr><td>1578 Definition (geometric l-face mapping)</td><td>92</td></tr><tr><td>1588 Definition (Lagrange nodes of P<sup>d</sup><sub>k</sub>)</td><td>97</td></tr><tr><td>1594 Definition (sub-vertices of Lagrange nodes of P<sup>d</sup><sub>k</sub>)</td><td>98</td></tr><tr><td>1608 Definition (Lagrange linear forms for P<sup>d</sup><sub>k</sub>)</td><td>102</td></tr><tr><td>1612 Definition (reference Lagrange linear forms for P<sup>d</sup><sub>k</sub>)</td><td>103</td></tr></table>
List of Lemmas

<table><tbody><tr><td>1358 Lemma (double induction by diagonal)</td><td>37</td></tr><tr><td>1360 Lemma (strong double induction)</td><td>38</td></tr><tr><td>1364 Lemma (properties of the binomial coefficient)</td><td>39</td></tr><tr><td>1366 Lemma (circular permutation)</td><td>40</td></tr><tr><td>1367 Lemma (transposition)</td><td>40</td></tr><tr><td>1368 Lemma (jump enumeration)</td><td>41</td></tr><tr><td>1369 Lemma (image of ker is included in ker)</td><td>41</td></tr><tr><td>1370 Lemma (image of ker is ker)</td><td>41</td></tr><tr><td>1372 Lemma (vector subspace is invariant by translation)</td><td>41</td></tr><tr><td>1373 Lemma (range of linear map is vector subspace)</td><td>41</td></tr><tr><td>1374 Lemma (injectivity or surjectivity and dimension implies bijectivity)</td><td>42</td></tr><tr><td>1375 Lemma (inverse of isomorphism is linear map)</td><td>42</td></tr><tr><td>1376 Lemma (free family of dim elements is basis)</td><td>42</td></tr><tr><td>1380 Lemma (origin is in affine subspace)</td><td>43</td></tr><tr><td>1381 Lemma (equivalent definition of affine subspace)</td><td>43</td></tr><tr><td>1382 Lemma (affine subspace is invariant by change of origin)</td><td>43</td></tr><tr><td>1383 Lemma (vector subspace is affine subspace)</td><td>43</td></tr><tr><td>1385 Lemma (affine subspace plus vector subspace is affine subspace)</td><td>43</td></tr><tr><td>1386 Lemma (closed under barycenter is affine subspace)</td><td>44</td></tr><tr><td>1387 Lemma (barycenter closure is affine subspace)</td><td>44</td></tr><tr><td>1390 Lemma (equivalent definition of affine map)</td><td>45</td></tr><tr><td>1391 Lemma (change of origin in affine map)</td><td>45</td></tr><tr><td>1393 Lemma (range of affine map is affine subspace)</td><td>46</td></tr><tr><td>1396 Lemma (space of affine maps)</td><td>46</td></tr><tr><td>1397 Lemma (output restriction of affine map)</td><td>46</td></tr><tr><td>1398 Lemma (affine submap)</td><td>46</td></tr><tr><td>1399 Lemma (equivalent definition of affine map (finite dimension))</td><td>47</td></tr><tr><td>1400 Lemma (affine map preserves barycenter)</td><td>47</td></tr><tr><td>1402 Lemma (affine map preserves isobarycenter)</td><td>47</td></tr><tr><td>1403 Lemma (affine maps are closed by composition)</td><td>47</td></tr><tr><td>1404 Lemma (continuous affine map is continuous linear map)</td><td>48</td></tr><tr><td>1405 Lemma (injective affine submap is zero linear kernel)</td><td>48</td></tr><tr><td>1406 Lemma (injective affine map is zero linear kernel)</td><td>48</td></tr><tr><td>1407 Lemma (surjective affine submap is full linear range)</td><td>48</td></tr><tr><td>1408 Lemma (surjective affine map is full linear range)</td><td>49</td></tr><tr><td>1409 Lemma (inverse of affine submap is affine submap)</td><td>49</td></tr><tr><td>1410 Lemma (inverse of affine map is affine map)</td><td>49</td></tr><tr><td>1412 Lemma (equivalent definition of affinely independent family)</td><td>50</td></tr><tr><td>1413 Lemma (affinely independent family of 2 elements)</td><td>50</td></tr><tr><td>1414 Lemma (affinely independent family is closed by sub-family)</td><td>51</td></tr><tr><td>1418 Lemma (P<sub>k</sub><sup>1</sup> is space of degree at most k)</td><td>51</td></tr><tr><td>1419 Lemma (monomials are free in P<sub>k</sub><sup>1</sup>)</td><td>51</td></tr><tr><td>1420 Lemma (dimension of P<sub>k</sub><sup>1</sup>)</td><td>52</td></tr><tr><td>1422 Lemma (product of two univariate polynomials)</td><td>52</td></tr><tr><td>1426 Lemma (injectivity implies unisolvence)</td><td>53</td></tr><tr><td>1427 Lemma (dimension of approximation space)</td><td>53</td></tr><tr><td>1428 Lemma (degrees of freedom are basis)</td><td>53</td></tr><tr><td>1435 Lemma (reference isobarycenter)</td><td>55</td></tr><tr><td>1436 Lemma (family of reference points is affinely independent)</td><td>55</td></tr></tbody></table>
<table>
  <tbody>
    <tr>
      <td>1438 Lemma (coordinates in reference simplex are smaller than 1)</td>
      <td>56</td>
    </tr>
    <tr>
      <td>1439 Lemma (nontrivial reference simplex)</td>
      <td>56</td>
    </tr>
    <tr>
      <td>1442 Lemma (coordinates in simplex are smaller than 1)</td>
      <td>56</td>
    </tr>
    <tr>
      <td>1443 Lemma (simplex of reference vertices is reference simplex)</td>
      <td>56</td>
    </tr>
    <tr>
      <td>1446 Lemma (cardinal of A<sub>k</sub><sup>1</sup>)</td>
      <td>57</td>
    </tr>
    <tr>
      <td>1449 Lemma (Lagrange polynomials is basis of P<sub>k</sub><sup>1</sup>)</td>
      <td>58</td>
    </tr>
    <tr>
      <td>1450 Lemma (decomposition of P<sub>k</sub><sup>1</sup> polynomial in Lagrange basis)</td>
      <td>58</td>
    </tr>
    <tr>
      <td>1452 Lemma (reference simplex is nontrivial in ℝ)</td>
      <td>59</td>
    </tr>
    <tr>
      <td>1454 Lemma (reference Lagrange nodes are distinct)</td>
      <td>59</td>
    </tr>
    <tr>
      <td>1455 Lemma (reference Lagrange basis of P<sub>k</sub><sup>1</sup>)</td>
      <td>59</td>
    </tr>
    <tr>
      <td>1457 Lemma (reference Lagrange linear forms for P<sub>k</sub><sup>1</sup> are linear)</td>
      <td>59</td>
    </tr>
    <tr>
      <td>1458 Lemma (reference Lagrange linear forms for P<sub>k</sub><sup>1</sup> are injective)</td>
      <td>59</td>
    </tr>
    <tr>
      <td>1459 Lemma (unisolvence of P<sub>k</sub><sup>1</sup> (reference))</td>
      <td>60</td>
    </tr>
    <tr>
      <td>1462 Lemma (properties of geometric mapping in dimension 1)</td>
      <td>60</td>
    </tr>
    <tr>
      <td>1463 Lemma (current simplex is nontrivial in ℝ)</td>
      <td>60</td>
    </tr>
    <tr>
      <td>1464 Lemma (current simplex is image of reference in ℝ)</td>
      <td>61</td>
    </tr>
    <tr>
      <td>1467 Lemma (Lagrange nodes are distinct)</td>
      <td>61</td>
    </tr>
    <tr>
      <td>1468 Lemma (Lagrange nodes of P<sub>k</sub><sup>1</sup> are images of reference)</td>
      <td>61</td>
    </tr>
    <tr>
      <td>1469 Lemma (Lagrange basis of P<sub>k</sub><sup>1</sup>)</td>
      <td>61</td>
    </tr>
    <tr>
      <td>1470 Lemma (Lagrange polynomials of P<sub>k</sub><sup>1</sup> are images of reference)</td>
      <td>61</td>
    </tr>
    <tr>
      <td>1471 Lemma (geometric mapping of P<sub>k</sub><sup>1</sup> is P<sub>k</sub><sup>1</sup>)</td>
      <td>62</td>
    </tr>
    <tr>
      <td>1473 Lemma (P<sub>k</sub><sup>1</sup> Lagrange linear forms are linear)</td>
      <td>62</td>
    </tr>
    <tr>
      <td>1474 Lemma (P<sub>k</sub><sup>1</sup> Lagrange linear forms are images of reference)</td>
      <td>62</td>
    </tr>
    <tr>
      <td>1476 Lemma (Lagrange linear forms for P<sub>k</sub><sup>1</sup> are injective)</td>
      <td>63</td>
    </tr>
    <tr>
      <td>1477 Lemma (unisolvence of P<sub>k</sub><sup>1</sup>)</td>
      <td>63</td>
    </tr>
    <tr>
      <td>1481 Lemma (length of multi-indices is additive)</td>
      <td>66</td>
    </tr>
    <tr>
      <td>1483 Lemma (factorial of multi-index is positive)</td>
      <td>66</td>
    </tr>
    <tr>
      <td>1485 Lemma (value of Kronecker delta of multi-indices)</td>
      <td>67</td>
    </tr>
    <tr>
      <td>1487 Lemma (multi-indices A<sub>k</sub><sup>d</sup> for d = 1 is A<sub>k</sub><sup>1</sup>)</td>
      <td>67</td>
    </tr>
    <tr>
      <td>1488 Lemma (indices are smaller than maximal length)</td>
      <td>67</td>
    </tr>
    <tr>
      <td>1489 Lemma (first C<sub>k</sub><sup>d</sup>)</td>
      <td>67</td>
    </tr>
    <tr>
      <td>1493 Lemma (slices of C<sub>k</sub><sup>d</sup>)</td>
      <td>68</td>
    </tr>
    <tr>
      <td>1494 Lemma (cardinal of slices of C<sub>k</sub><sup>d</sup>)</td>
      <td>69</td>
    </tr>
    <tr>
      <td>1495 Lemma (cardinal of C<sub>k</sub><sup>d</sup>)</td>
      <td>69</td>
    </tr>
    <tr>
      <td>1496 Lemma (C<sub>k</sub><sup>d</sup> are layers of A<sub>k</sub><sup>d</sup>)</td>
      <td>70</td>
    </tr>
    <tr>
      <td>1497 Lemma (first multi-indices A<sub>k</sub><sup>d</sup>)</td>
      <td>70</td>
    </tr>
    <tr>
      <td>1498 Lemma (cardinal of A<sub>k</sub><sup>d</sup>)</td>
      <td>70</td>
    </tr>
    <tr>
      <td>1500 Lemma (cardinal of C<sub>k</sub><sup>d</sup> and A<sub>k</sub><sup>d-1</sup>)</td>
      <td>70</td>
    </tr>
    <tr>
      <td>1501 Lemma (cardinal of A<sub>k,i</sub><sup>d</sup> and A<sub>k</sub><sup>d-1</sup>)</td>
      <td>71</td>
    </tr>
    <tr>
      <td>1504 Lemma (monomial in d variables for d = 1 is monomial of a single variable)</td>
      <td>72</td>
    </tr>
    <tr>
      <td>1506 Lemma (polynomial space P<sub>k</sub><sup>d</sup> for d = 1 is P<sub>k</sub><sup>1</sup>)</td>
      <td>72</td>
    </tr>
    <tr>
      <td>1507 Lemma (P<sub>k</sub><sup>d</sup> is vector space)</td>
      <td>72</td>
    </tr>
    <tr>
      <td>1508 Lemma (P<sub>k</sub><sup>d</sup> is nondecreasing sequence in k)</td>
      <td>72</td>
    </tr>
    <tr>
      <td>1509 Lemma (constant and affine spaces P<sub>0</sub><sup>d</sup> and P<sub>1</sub><sup>d</sup>)</td>
      <td>73</td>
    </tr>
    <tr>
      <td>1511 Lemma (values of degree of polynomial)</td>
      <td>73</td>
    </tr>
    <tr>
      <td>1512 Lemma (monomials of C<sub>k</sub><sup>d</sup> have degree k)</td>
      <td>73</td>
    </tr>
    <tr>
      <td>1513 Lemma (P<sub>k</sub><sup>d</sup> is space of degree at most k)</td>
      <td>73</td>
    </tr>
    <tr>
      <td>1514 Lemma (product of monomials)</td>
      <td>73</td>
    </tr>
    <tr>
      <td>1515 Lemma (product of monomial and polynomial)</td>
      <td>74</td>
    </tr>
    <tr>
      <td>1516 Lemma (product of two polynomials)</td>
      <td>74</td>
    </tr>
    <tr>
      <td>1517 Lemma (partial derivative of monomials)</td>
      <td>74</td>
    </tr>
    <tr>
      <td>1519 Lemma (partial derivative is linear)</td>
      <td>75</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>1520 Lemma (partial derivative of 0)</td>
      <td>75</td>
    </tr>
    <tr>
      <td>1521 Lemma (derivating more than degree is 0)</td>
      <td>75</td>
    </tr>
    <tr>
      <td>1522 Lemma (partial derivative of monomials at 0)</td>
      <td>75</td>
    </tr>
    <tr>
      <td>1523 Lemma (monomials are free in P<sup>d</sup><sub>k</sub>)</td>
      <td>75</td>
    </tr>
    <tr>
      <td>1524 Lemma (monomials are a basis of P<sup>d</sup><sub>k</sub>)</td>
      <td>75</td>
    </tr>
    <tr>
      <td>1525 Lemma (dimension of P<sup>d</sup><sub>k</sub>)</td>
      <td>76</td>
    </tr>
    <tr>
      <td>1526 Lemma (isomorphism between P<sup>d</sup><sub>0</sub> and P<sup>d-1</sup><sub>0</sub>)</td>
      <td>76</td>
    </tr>
    <tr>
      <td>1529 Lemma (decomposition of P<sup>d</sup><sub>k</sub>)</td>
      <td>76</td>
    </tr>
    <tr>
      <td>1531 Lemma (isomorphism between P<sup>d</sup><sub>k</sub> and P<sup>d-1</sup><sub>k</sub> × P<sup>d</sup><sub>k-1</sub>)</td>
      <td>77</td>
    </tr>
    <tr>
      <td>1533 Lemma (P<sup>d</sup><sub>k</sub> is nondecreasing sequence in d)</td>
      <td>78</td>
    </tr>
    <tr>
      <td>1534 Lemma (expression of P<sup>d</sup><sub>k</sub> as polynomial of x<sub>d</sub>)</td>
      <td>78</td>
    </tr>
    <tr>
      <td>1537 Lemma (product of two polynomials (alternate proof))</td>
      <td>79</td>
    </tr>
    <tr>
      <td>1538 Lemma (product of polynomials)</td>
      <td>80</td>
    </tr>
    <tr>
      <td>1539 Lemma (affine mapping of monomials is P<sup>l</sup><sub>k</sub>)</td>
      <td>80</td>
    </tr>
    <tr>
      <td>1540 Lemma (affine mapping of P<sup>d</sup><sub>k</sub> is P<sup>l</sup><sub>k</sub>)</td>
      <td>80</td>
    </tr>
    <tr>
      <td>1542 Lemma (reference Lagrange polynomials of P<sup>d</sup><sub>1</sub> for d = 1 are reference Lagrange polynomials of P<sup>1</sup><sub>k</sub> for k = 1)</td>
      <td>81</td>
    </tr>
    <tr>
      <td>1543 Lemma (reference Lagrange polynomials is basis of P<sup>d</sup><sub>1</sub>)</td>
      <td>81</td>
    </tr>
    <tr>
      <td>1545 Lemma (differential of reference Lagrange polynomials)</td>
      <td>81</td>
    </tr>
    <tr>
      <td>1548 Lemma (geometric mapping for d = 1 is geometric mapping in dimension 1)</td>
      <td>82</td>
    </tr>
    <tr>
      <td>1549 Lemma (reference geometric mapping is identity)</td>
      <td>82</td>
    </tr>
    <tr>
      <td>1550 Lemma (properties of geometric mapping)</td>
      <td>83</td>
    </tr>
    <tr>
      <td>1551 Lemma (differential of geometric mapping)</td>
      <td>84</td>
    </tr>
    <tr>
      <td>1552 Lemma (Lagrange polynomials of P<sup>d</sup><sub>1</sub>)</td>
      <td>84</td>
    </tr>
    <tr>
      <td>1553 Lemma (Lagrange polynomials of reference vertices are reference Lagrange polynomials of P<sup>d</sup><sub>1</sub>)</td>
      <td>84</td>
    </tr>
    <tr>
      <td>1554 Lemma (Lagrange polynomials is basis of P<sup>d</sup><sub>1</sub>)</td>
      <td>85</td>
    </tr>
    <tr>
      <td>1555 Lemma (decomposition of P<sup>d</sup><sub>1</sub> polynomial in Lagrange basis)</td>
      <td>85</td>
    </tr>
    <tr>
      <td>1556 Lemma (differential of Lagrange polynomials P<sup>d</sup><sub>1</sub>)</td>
      <td>85</td>
    </tr>
    <tr>
      <td>1557 Lemma (nontrivial simplex)</td>
      <td>86</td>
    </tr>
    <tr>
      <td>1559 Lemma (barycentric coordinate)</td>
      <td>86</td>
    </tr>
    <tr>
      <td>1560 Lemma (Lagrange polynomials of P<sup>d</sup><sub>1</sub> are barycentric coordinate)</td>
      <td>87</td>
    </tr>
    <tr>
      <td>1561 Lemma (decomposition of P<sup>d</sup><sub>1</sub> polynomial with barycentric coordinates)</td>
      <td>87</td>
    </tr>
    <tr>
      <td>1563 Lemma (equivalent definition of face hyperplane)</td>
      <td>88</td>
    </tr>
    <tr>
      <td>1564 Lemma (reference face hyperplane)</td>
      <td>89</td>
    </tr>
    <tr>
      <td>1565 Lemma (face hyperplane is image of reference face hyperplane)</td>
      <td>89</td>
    </tr>
    <tr>
      <td>1567 Lemma (hyperface is included in face hyperplane)</td>
      <td>90</td>
    </tr>
    <tr>
      <td>1569 Lemma (equivalent definition of l-face affine space)</td>
      <td>91</td>
    </tr>
    <tr>
      <td>1570 Lemma (d-face affine space is full space)</td>
      <td>91</td>
    </tr>
    <tr>
      <td>1571 Lemma (0-face affine space is vertex)</td>
      <td>91</td>
    </tr>
    <tr>
      <td>1573 Lemma (l-face is included in l-face affine space)</td>
      <td>91</td>
    </tr>
    <tr>
      <td>1574 Lemma (d-face is simplex)</td>
      <td>91</td>
    </tr>
    <tr>
      <td>1576 Lemma ((d-1)-face is hyperface)</td>
      <td>92</td>
    </tr>
    <tr>
      <td>1579 Lemma (geometric d-face mapping is geometric mapping)</td>
      <td>92</td>
    </tr>
    <tr>
      <td>1581 Lemma (properties of geometric l-face mapping)</td>
      <td>93</td>
    </tr>
    <tr>
      <td>1582 Lemma (geometric l-face mapping of P<sup>d</sup><sub>k</sub> is P<sup>l</sup><sub>k</sub>)</td>
      <td>94</td>
    </tr>
    <tr>
      <td>1583 Lemma (geometric mapping of P<sup>d</sup><sub>k</sub> is P<sup>l</sup><sub>k</sub>)</td>
      <td>94</td>
    </tr>
    <tr>
      <td>1584 Lemma (geometric hyperface mapping)</td>
      <td>94</td>
    </tr>
    <tr>
      <td>1585 Lemma (hyperface geometric mapping of P<sup>d</sup><sub>k</sub> is P<sup>d-1</sup><sub>k</sub>)</td>
      <td>95</td>
    </tr>
    <tr>
      <td>1586 Lemma (geometric mapping with permutation)</td>
      <td>96</td>
    </tr>
    <tr>
      <td>1589 Lemma (Lagrange nodes of P<sup>d</sup><sub>k</sub> for d = 1 are Lagrange nodes of P<sup>1</sup><sub>k</sub>)</td>
      <td>97</td>
    </tr>
    <tr>
      <td>1590 Lemma (number of Lagrange nodes of P<sup>d</sup><sub>k</sub>)</td>
      <td>97</td>
    </tr>
    <tr>
      <td>1591 Lemma (barycentric coordinates of Lagrange nodes of P<sup>d</sup><sub>k</sub>)</td>
      <td>98</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>1592 Lemma (vertices are Lagrange nodes of P<sup>d</sup><sub>k</sub>)</td>
      <td>98</td>
    </tr>
    <tr>
      <td>1593 Lemma (Lagrange nodes of P<sup>d</sup><sub>1</sub> are vertices)</td>
      <td>98</td>
    </tr>
    <tr>
      <td>1595 Lemma (equivalent definition of sub-vertices of Lagrange nodes of P<sup>d</sup><sub>k</sub>)</td>
      <td>99</td>
    </tr>
    <tr>
      <td>1597 Lemma (sub-vertices are affinely independent)</td>
      <td>99</td>
    </tr>
    <tr>
      <td>1598 Lemma (P<sup>d</sup><sub>k-1</sub> sub-nodes of sub-vertices are some nodes of P<sup>d</sup><sub>k-1</sub>)</td>
      <td>100</td>
    </tr>
    <tr>
      <td>1599 Lemma (reference Lagrange nodes of P<sup>d</sup><sub>k</sub>)</td>
      <td>100</td>
    </tr>
    <tr>
      <td>1600 Lemma (reference Lagrange nodes of P<sup>d</sup><sub>k</sub> for d = 1 are reference Lagrange nodes of P<sup>1</sup><sub>k</sub>)</td>
      <td>100</td>
    </tr>
    <tr>
      <td>1601 Lemma (equivalent definition of reference Lagrange nodes of P<sup>d</sup><sub>k</sub>)</td>
      <td>101</td>
    </tr>
    <tr>
      <td>1602 Lemma (number of reference Lagrange nodes of P<sup>d</sup><sub>k</sub>)</td>
      <td>101</td>
    </tr>
    <tr>
      <td>1604 Lemma (Lagrange nodes of P<sup>d</sup><sub>k</sub> are image of reference)</td>
      <td>101</td>
    </tr>
    <tr>
      <td>1605 Lemma (face hyperplanes of Lagrange nodes of P<sup>d</sup><sub>k</sub>)</td>
      <td>101</td>
    </tr>
    <tr>
      <td>1607 Lemma (image of nodes by geometric hyperface mapping)</td>
      <td>102</td>
    </tr>
    <tr>
      <td>1609 Lemma (Lagrange linear forms for P<sup>d</sup><sub>k</sub> for d=1 are Lagrange linear forms for P<sup>1</sup><sub>k</sub>)</td>
      <td>102</td>
    </tr>
    <tr>
      <td>1610 Lemma (Lagrange linear forms of P<sup>d</sup><sub>k</sub> are linear)</td>
      <td>103</td>
    </tr>
    <tr>
      <td>1611 Lemma (cardinal of Lagrange linear forms of P<sup>d</sup><sub>k</sub>)</td>
      <td>103</td>
    </tr>
    <tr>
      <td>1613 Lemma (reference Lagrange linear forms for P<sup>d</sup><sub>k</sub> for d = 1 are reference Lagrange linear forms for P<sup>1</sup><sub>k</sub>)</td>
      <td>103</td>
    </tr>
    <tr>
      <td>1614 Lemma (Lagrange linear forms of P<sup>d</sup><sub>k</sub> are images of reference)</td>
      <td>103</td>
    </tr>
    <tr>
      <td>1615 Lemma (Lagrange linear forms for P<sup>d</sup><sub>0</sub> are injective)</td>
      <td>103</td>
    </tr>
    <tr>
      <td>1617 Lemma (unisolvence of P<sup>d</sup><sub>0</sub>)</td>
      <td>104</td>
    </tr>
    <tr>
      <td>1618 Lemma (decomposition of P<sup>d</sup><sub>1</sub> polynomial with σ<sub>α</sub>)</td>
      <td>104</td>
    </tr>
    <tr>
      <td>1619 Lemma (Lagrange linear forms for P<sup>d</sup><sub>1</sub> are injective)</td>
      <td>104</td>
    </tr>
    <tr>
      <td>1620 Lemma (unisolvence of P<sup>d</sup><sub>1</sub>)</td>
      <td>104</td>
    </tr>
    <tr>
      <td>1621 Lemma (factorization of zero polynomial on last reference hyperplane)</td>
      <td>104</td>
    </tr>
    <tr>
      <td>1623 Lemma (factorization of zero polynomial on hyperplane P<sup>d</sup><sub>k</sub>)</td>
      <td>105</td>
    </tr>
    <tr>
      <td>1625 Lemma (Lagrange linear forms for P<sup>d</sup><sub>k</sub> are injective)</td>
      <td>105</td>
    </tr>
    <tr>
      <td>1628 Lemma (face unisolvence of P<sup>d</sup><sub>k</sub>)</td>
      <td>107</td>
    </tr>
    <tr>
      <td>1630 Lemma (Log P<sup>d</sup><sub>k</sub> Lagrange finite element for d = 1 is Lagrange finite element on a current segment)</td>
      <td>108</td>
    </tr>
    <tr>
      <td>1632 Lemma (Log P̂<sup>d</sup><sub>k</sub> reference Lagrange finite element for d = 1 is Lagrange finite element on the reference segment)</td>
      <td>108</td>
    </tr>
  </tbody>
</table>
List of Theorems

<table><tr><td>1460 Theorem (<sup>LAG</sup>P<sup>1</sup><sub>k</sub> reference Lagrange finite element)</td><td>60</td></tr><tr><td>1478 Theorem (<sup>LAG</sup>P<sup>1</sup><sub>k</sub> Lagrange finite element)</td><td>63</td></tr><tr><td>1626 Theorem (unisolvence of P<sup>d</sup><sub>k</sub>)</td><td>107</td></tr><tr><td>1629 Theorem (<sup>LAG</sup>P<sup>d</sup><sub>k</sub> Lagrange finite element)</td><td>108</td></tr><tr><td>1631 Theorem (<sup>LAG</sup>P<sup>d</sup><sub>k</sub> reference Lagrange finite element)</td><td>108</td></tr></table>
# Appendix B

## The proof cites explicitly...

This appendix gathers the explicit citations of the statements listed in Appendix A that appear in the proof of each result (lemmas and theorems). Statements from [14, 15] are anonymized.

The dual graph is described in Appendix C.

Printing is not advised!

**The proof of Lemma 1358 (double induction by diagonal)**

has no explicit citation.

**The proof of Lemma 1360 (strong double induction)**

has no explicit citation.

**The proof of Lemma 1364 (properties of the binomial coefficient)**

cites explicitly:

Definition 1362 (binomial coefficient).

**The proof of Lemma 1366 (circular permutation)**

has no explicit citation.

**The proof of Lemma 1367 (transposition)**

has no explicit citation.

**The proof of Lemma 1368 (jump enumeration)**

has no explicit citation.

**The proof of Lemma 1369 (image of $ker$ is included in $ker$)**

cites explicitly:

Statement(s) from [14].

**The proof of Lemma 1370 (image of $ker$ is $ker$)**

cites explicitly:

Statement(s) from [14],

Lemma 1369 (image of $ker$ is included in $ker$).

**The proof of Lemma 1372 (vector subspace is invariant by translation)**

cites explicitly:

Statement(s) from [14].

**The proof of Lemma 1373 (range of linear map is vector subspace)**

cites explicitly:

Statement(s) from [14].
**The proof of Lemma 1374 (injectivity or surjectivity and dimension implies bijectivity)**

cites explicitly:

Statement(s) from [14].

**The proof of Lemma 1375 (inverse of isomorphism is linear map)**

cites explicitly:

Statement(s) from [14].

**The proof of Lemma 1376 (free family of dim elements is basis)**

has no explicit citation.

**The proof of Lemma 1380 (origin is in affine subspace)**

cites explicitly:

Statement(s) from [14],

Definition 1379 (affine subspace).

**The proof of Lemma 1381 (equivalent definition of affine subspace)**

cites explicitly:

Statement(s) from [14],

Lemma 1372 (vector subspace is invariant by translation),

Definition 1379 (affine subspace).

**The proof of Lemma 1382 (affine subspace is invariant by change of origin)**

cites explicitly:

Statement(s) from [14],

Lemma 1372 (vector subspace is invariant by translation),

Definition 1379 (affine subspace),

Lemma 1381 (equivalent definition of affine subspace).

**The proof of Lemma 1383 (vector subspace is affine subspace)**

cites explicitly:

Lemma 1372 (vector subspace is invariant by translation),

Lemma 1381 (equivalent definition of affine subspace).

**The proof of Lemma 1385 (affine subspace plus vector subspace is affine subspace)**

cites explicitly:

Statement(s) from [14],

Definition 1379 (affine subspace),

Lemma 1381 (equivalent definition of affine subspace).

**The proof of Lemma 1386 (closed under barycenter is affine subspace)**

cites explicitly:

Statement(s) from [14],

Definition 1379 (affine subspace),

Lemma 1381 (equivalent definition of affine subspace).

**The proof of Lemma 1387 (barycenter closure is affine subspace)**

cites explicitly:

Statement(s) from [14],

Definition 1379 (affine subspace).

**The proof of Lemma 1390 (equivalent definition of affine map)**

cites explicitly:

Definition 1379 (affine subspace),

Lemma 1381 (equivalent definition of affine subspace),

Definition 1388 (affine map).
**The proof of Lemma 1391 (change of origin in affine map)**

cites explicitly:
* Statement(s) from [14],
* Definition 1379 (affine subspace),
* Lemma 1381 (equivalent definition of affine subspace),
* Definition 1388 (affine map).

**The proof of Lemma 1393 (range of affine map is affine subspace)**

cites explicitly:
* Lemma 1373 (range of linear map is vector subspace),
* Definition 1379 (affine subspace),
* Definition 1388 (affine map).

**The proof of Lemma 1396 (space of affine maps)**

cites explicitly:
* Statement(s) from [14],
* Definition 1395 (set of affine maps).

**The proof of Lemma 1397 (output restriction of affine map)**

cites explicitly:
* Lemma 1382 (affine subspace is invariant by change of origin),
* Lemma 1393 (range of affine map is affine subspace),
* Definition 1395 (set of affine maps).

**The proof of Lemma 1398 (affine submap)**

cites explicitly:
* Definition 1379 (affine subspace),
* Definition 1388 (affine map),
* Lemma 1391 (change of origin in affine map),
* Definition 1395 (set of affine maps),
* Lemma 1397 (output restriction of affine map).

**The proof of Lemma 1399 (equivalent definition of affine map (finite dimension))**

cites explicitly:
* Definition 1388 (affine map).

**The proof of Lemma 1400 (affine map preserves barycenter)**

cites explicitly:
* Statement(s) from [14],
* Definition 1388 (affine map).

**The proof of Lemma 1402 (affine map preserves isobarycenter)**

cites explicitly:
* Statement(s) from [14],
* Lemma 1400 (affine map preserves barycenter),
* Definition 1401 (isobarycenter).

**The proof of Lemma 1403 (affine maps are closed by composition)**

cites explicitly:
* Statement(s) from [14],
* Definition 1388 (affine map),
* Lemma 1390 (equivalent definition of affine map),
* Definition 1395 (set of affine maps).

**The proof of Lemma 1404 (continuous affine map is continuous linear map)**

cites explicitly:
* Statement(s) from [14],
* Lemma 1390 (equivalent definition of affine map).
**The proof of Lemma 1405 (injective affine submap is zero linear kernel)**

cites explicitly:
* Statement(s) from [14],
* Lemma 1398 (affine submap).

**The proof of Lemma 1406 (injective affine map is zero linear kernel)**

cites explicitly:
* Lemma 1405 (injective affine submap is zero linear kernel).

**The proof of Lemma 1407 (surjective affine submap is full linear range)**

cites explicitly:
* Statement(s) from [14],
* Definition 1379 (affine subspace),
* Lemma 1381 (equivalent definition of affine subspace),
* Lemma 1382 (affine subspace is invariant by change of origin),
* Definition 1388 (affine map),
* Lemma 1390 (equivalent definition of affine map).

**The proof of Lemma 1408 (surjective affine map is full linear range)**

cites explicitly:
* Lemma 1407 (surjective affine submap is full linear range).

**The proof of Lemma 1409 (inverse of affine submap is affine submap)**

cites explicitly:
* Statement(s) from [14],
* Lemma 1375 (inverse of isomorphism is linear map),
* Definition 1379 (affine subspace),
* Lemma 1381 (equivalent definition of affine subspace),
* Definition 1388 (affine map),
* Lemma 1390 (equivalent definition of affine map),
* Lemma 1405 (injective affine submap is zero linear kernel),
* Lemma 1407 (surjective affine submap is full linear range).

**The proof of Lemma 1410 (inverse of affine map is affine map)**

cites explicitly:
* Lemma 1409 (inverse of affine submap is affine submap).

**The proof of Lemma 1412 (equivalent definition of affinely independent family)**

cites explicitly:
* Statement(s) from [14],
* Definition 1411 (affinely independent family).

**The proof of Lemma 1413 (affinely independent family of 2 elements)**

cites explicitly:
* Definition 1411 (affinely independent family).

**The proof of Lemma 1414 (affinely independent family is closed by sub-family)**

cites explicitly:
* Definition 1411 (affinely independent family),
* Lemma 1412 (equivalent definition of affinely independent family).

**The proof of Lemma 1418 ($\mathbb{P}_k^1$ is space of degree at most $k$)**

cites explicitly:
* Lemma 1396 (space of affine maps),
* Lemma 1399 (equivalent definition of affine map (finite dimension)),
* Definition 1416 (monomial of a single variable),
* Definition 1417 (polynomial space $\mathbb{P}_k^1$).
**The proof of Lemma 1419 (monomials are free in $\mathbb{P}_k^1$)**
has no explicit citation.

**The proof of Lemma 1420 (dimension of $\mathbb{P}_k^1$)**

cites explicitly:
Lemma 1376 (free family of dim elements is basis),
Definition 1417 (polynomial space $\mathbb{P}_k^1$),
Lemma 1419 (monomials are free in $\mathbb{P}_k^1$).

**The proof of Lemma 1422 (product of two univariate polynomials)**
has no explicit citation.

**The proof of Lemma 1426 (injectivity implies unisolvence)**

cites explicitly:
Lemma 1374 (injectivity or surjectivity and dimension implies bijectivity),
Definition 1424 (finite element triple).

**The proof of Lemma 1427 (dimension of approximation space)**
has no explicit citation.

**The proof of Lemma 1428 (degrees of freedom are basis)**

cites explicitly:
Lemma 1376 (free family of dim elements is basis),
Lemma 1427 (dimension of approximation space).

**The proof of Lemma 1435 (reference isobarycenter)**

cites explicitly:
Definition 1401 (isobarycenter),
Definition 1434 (family of reference points).

**The proof of Lemma 1436 (family of reference points is affinely independent)**

cites explicitly:
Definition 1365 (canonic families),
Definition 1411 (affinely independent family),
Definition 1434 (family of reference points).

**The proof of Lemma 1438 (coordinates in reference simplex are smaller than 1)**

cites explicitly:
Definition 1437 (reference simplex).

**The proof of Lemma 1439 (nontrivial reference simplex)**

cites explicitly:
Definition 1437 (reference simplex).

**The proof of Lemma 1442 (coordinates in simplex are smaller than 1)**

cites explicitly:
Definition 1440 (simplex).

**The proof of Lemma 1443 (simplex of reference vertices is reference simplex)**

cites explicitly:
Definition 1434 (family of reference points),
Definition 1437 (reference simplex),
Definition 1440 (simplex).

**The proof of Lemma 1446 (cardinal of $A_k^1$)**

cites explicitly:
Lemma 1364 (properties of the binomial coefficient),
Definition 1445 (multi-indices $A_k^1$).