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# Length 2 Variables of A[x, y] and Transfer

Eric Edo, Stéphane Vénéreau

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Eric Edo, Stéphane Vénéreau. Length 2 Variables of A[x, y] and Transfer. Annales Polonici Mathe-
matici, 2001, 76 (1-2), pp.67-76. hal-04815465

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# Length 2 Variables of $A[x, y]$ and Transfer.

Eric Edo<fnref n="1" /> and Stéphane Vénéreau<fnref n="2" />

**Abstract.** We construct and study length 2 variables of $A[x, y]$ (A is a commutative ring). If A is an integral domain, we determine among these variables those which are tame. If A is a UFD, we prove that these variables are all stably tame. We apply this construction to show that some polynomials of $A[x_1, \cdots, x_n]$ are variables using transfer.

## 1 Introduction.

Let $A$ be a commutative ring, the automorphisms group $GA_n(A)$ is anti-isomorphic to $AutA_A^n$ and has an obvious geometric significance. A naïve question about automorphisms is: “What do they look like ?” This is already a difficult problem for $n$ equal 2 and we focus on this case. The simplest automorphisms belong to the tree subgroups $Af_2(A)$, $BA_2(A)$ and $N_n(A)$ (see Notations 1). When $A$ is a field the situation is well understood thanks to Jung-van der Kulk’s theorem (see Theorem 1): all automorphisms are tame *i.e.* belong to the subgroup generated by $Af_2(A)$ and $BA_2(A)$. But there exist non tame automorphisms as soon as $A$ is no longer a field. Since $n$ equal 2 the study of automorphisms boils down to the study of variables (see Corollary 2). When $\mathbb{Q} \subset A$ a general result (see Theorem 2) gives a criterion for a polynomial to be a variable. Without assumption on $A$, we define the length of a variable (see Notation 3). Length 1 variables were described by Russell and Sathaye (see Theorem 3). The study of length 2 variables has begun only recently in the work of Drensky and Yu [DY], but they use characteristic zero techniques; in fact they suppose $A = K[z]$ with $K$ a field of characteristic zero and in this case the general criterion can be applied. We give a construction of length 2 variables without any assumption on $A$ (see Theorem 4) and we study these variables. If $A$ is an integral domain, we

<footnote n="1">¹Département de Mathématiques pures, Université Bordeaux I, 351 cours de la Libération, 33405 Talence Cedex, France. E-mail: edo@math.u-bordeaux.fr</footnote>

<footnote n="2">²Institut Fourier, Université Joseph Fourier, B. P. 53, 38041 Grenoble Cedex 9, France. E-mail: venereau@ujf-grenoble.fr</footnote>
determine among these variables those which are tame (see Proposition 3) using Jung-van der Kulk's theorem. If A is a UFD, we prove that these variables are all stably tame (see Theorem 5). This result was obtained with the help of D. Wright and is a strong generalization of M. Smith's result (see [S]). Finally, we apply this construction to show that some polynomials of A[x₁, ..., xₙ] are variables using transfer (see Theorem 7 and Theorem 8).

## 2 Notations and background.

Throughout this paper, A and k denote commutative rings, A^x is the set of nonzerodivisors of A and qtA denotes the total quotient ring of A i.e. qtA = (A^x)^{-1}A.

**Notations 1** We use the following classical notations:

$GA_n(A) = Aut_A A[x_1, ..., x_n]$ is the automorphisms group of the A-algebra $A[x_1, ..., x_n]$,

$Af_n(A)$ is the affine automorphisms subgroup,

$BA_n(A)$ is the triangular automorphisms subgroup ($\beta \in GA_n(A)$) is triangular if $\beta(x_i) = a_i x_i + P_i(x_{i+1}, ..., x_n)$ where $a_i$ are units in A and $P_i \in A[x_{i+1}, ..., x_n]$ for $1 \le i \le n$),

$N_n(A)$ is the subgroup of nilpotency (($\beta \in N_n(A)$ if $\beta(x_i) = x_i + P_i$ with $P_i$ nilpotent elements in $A[x_1, ..., x_n]$).

$TA_n(A)$ is the tame automorphisms subgroup (i.e. the subgroup of $GA_n(A)$ generated by $Af_n(A)$, $BA_n(A)$ and $N_n(A)$),

$ST_n(A)$ the stably tame automorphisms subgroup ($\sigma \in GA_n(A)$ is stably tame if there exists $m \ge 1$ such that $\sigma_m \in TA_{n+m}(A)$ where $\sigma_m \in GA_{n+m}(A)$ is defined by $\sigma_m(x_i) = \sigma(x_i)$ if $1 \le i \le n$ and $\sigma_m(x_i) = x_i$ if $n+1 \le i \le n+m$). If $n \le 3$ we will use the following notations: $x = x_1$, $y = x_2$ and $z = x_3$.

The following result is well-known (see for exemple [N] or [MW]):

**Theorem 1 (Jung, van der Kulk)** If A is a field then

$$GA_2(A) = TA_2(A) = Af_2(A) * BA_2(A)$$

where * is the amalgamated product of $Af_2(A)$ and $BA_2(A)$ along their intersection.

**Notations 2** We introduce new notations:

$$TA_2^n(A) = \{\sigma \in TA_2(A); \sigma = b_n a_n \dots b_1 a_1 b_0; b_i \in BA_2(A), a_i \in Af_2(A)\},$$
$$GA_2^n(A) = GA_2(A) \cap TA_2^n(qtA).$$

The Jung-van der Kulk theorem has the following useful consequence:

**Corollary 1** If A is a integral domain then

$$GA_2(A) = \bigcup_{n \in \mathbb{N}} GA_2^n(A)$$

and

$$GA_2^n(A) \cap TA_2(A) = TA_2^n(A).$$

**Proof** Because A is a integral domain, qtA is a field and we can apply the theorem in qtA. The first equation follows from $GA_2(qtA) = TA_2(qtA)$. The inclusion $TA_2^n(A) \subset GA_2^n(A) \cap TA_2(A)$ is obvious, the opposite one follows from $TA_2(A) = Af_2(A) * BA_2(A)$ and the two inclusions $Af_2(A) \setminus BA_2(A) \subset Af_2(qtA) \setminus BA_2(qtA)$ and $BA_2(A) \setminus Af_2(A) \subset BA_2(qtA) \setminus Af_2(qtA)$.

One can easily describe $GA_1(A)$ (cf. [N] Proposition 3.1):

**Proposition 1** We have:

$$GA_1(A) = \{ax + b(x) + c, a \text{ unit in } A, b \text{ nilpotent in } A[x], c \in A\}$$

**Corollary 2** Let $\sigma, \sigma' \in GA_2(A)$ if $\sigma(y) = \sigma'(y)$ then $\sigma^{-1}\sigma' \in BA_2(A)N_2(A)$.

This corollary shows that all the information about appearance and tameness of an automorphism of $A[x, y]$ is contained in one of his two components i.e. in a variable.

**Notations 3** We define subsets of $A[x, y]$:

$$TV_2(A) = \{F \in A[x, y]; \exists \sigma \in TA_2(A); \sigma(y) = F\},$$

$$TV_2^n(A) = \{F \in A[x, y]; \exists \sigma \in TA_2^n(A); \sigma(y) = F\},$$

$$VA_2(A) = \{F \in A[x, y]; \exists \sigma \in GA_2(A); \sigma(y) = F\},$$

$$VA_2^n(A) = \{F \in A[x, y]; \exists \sigma \in GA_2^n(A); \sigma(y) = F\},$$

$$SV_2(A) = \{F \in A[x, y]; \exists \sigma \in ST_2(A); \sigma(y) = F\}.$$

**Definitions** An element of $TV_2(A)$ (resp. $VA_2(A)$, resp. $SV_2(A)$) is called a tame variable (resp. a variable, resp. a stably tame variable). A length $n$ variable is an element of $VA_2^n(A) \setminus VA_2^{n-1}(A)$.

Here is a powerful criterion for proving that some polynomials are variables:
**Theorem 2** Suppose $\mathbb{Q} \subset A$ and let $F \in A[x, y]$. Then $F \in VA_2(A)$ if and only if the two following assumptions hold:

i) $F \in VA_2(\text{qtA})$,

ii) $1 \in (\partial_x F, \partial_y F)$.

This theorem is based on the theory of locally nilpotent derivations developed recently by Daigle and Freudenburg when $A$ is a UFD, see [DF] Proposition 2.3 and Theorem 2.5, Bhatwadekar and Dutta when $A$ is a (normal) noetherian integral domain see [BD] Theorem 4.7 and Berson, van den Essen and Maubach in the general situation (i.e. when $\mathbb{Q} \subset A$) see [BEM] Theorem 3.7.

If $\mathbb{Q} \not\subset A$ then Theorem 2 is not true as soon as $A$ is not a field.

If $\text{char}(A) = p > 0$, let $q \in A^\times$ and suppose $q$ is not a unit in $A$, we consider $F = qx + y + y^p \in VA_2(\text{qtA})$. Let $\mathcal{M}$ a maximal ideal such that $q \in \mathcal{M}$, we have $F = y(1 + y^{p-1}) \text{ mod } \mathcal{M}$, hence $\bar{F} \not\in VA_2(A/\mathcal{M})$ and $F \not\in VA_2(A)$. Nevertheless assumptions i) and ii) hold.

If $\text{char}(A) = 0$, let $q \in A^\times$ and suppose $q$ is an integer and is not a unit in $A$, we consider $F = qx + y + y^q$ and we conclude as above.

On the other hand we have (see [R]) this old result (without the assumption $\mathbb{Q} \subset A$) which describes $VA_2^1(A)$:

**Theorem 3 (Russell, Sathaye)** Let $F \in A[x, y]$. Then $F \in VA_2^1(A)$ if and only if $F(x, y) = px + \sum g_i y^i$ with $p \in A^\times$ and $\sum g_i y^i \in A[y]$ such that $g_1$ (resp. $g_i$ ($i \ge 2$)) is a unit (resp. are nilpotent) mod $pA$.

**Remark** If $\mathbb{Q} \subset A$ then Theorem 3 follows from Theorem 2.

Corollary 1 gives:

**Proposition 2** If $A$ is a integral domain then
$$VA_2^1(A) \cap TV_2(A) = \{px + \sum g_i y^i \in VA_2^1(A); \forall i \ge 2 \ g_i = 0 \ \text{mod } pA\}.$$

**Remark** The Nagata polynomial is $N = z^2x + y + zy^2 \in A[x, y]$ with $A = k[z]$. Theorem 3 and Proposition 2 show that $N \in VA_2^1(A) \setminus TV_2(A)$ which is a result due to Nagata (see [N]).

# 3 Length 2 variables of $A[x, y]$

**Notation** We denote by $LV_2^2(A)$ the set of polynomials $y+H(px+G(y))$ with $p \in A^\times$, $G(y) = \sum g_i y^i \in A[y]$ and $H(y) = \sum h_i y^i \in A[y]$ such that $h_i$ is nilpotent mod $pA$ for $i \ge 1$.
**Theorem 4** We have the following inclusion: $LV_2^2(A) \subset VA_2^2(A)$.

**Proof** Let $p \in A^\times$, $G(y) = \sum g_i y^i \in A[y]$ and $H(y) = \sum h_i y^i \in A[y]$ such that $h_i$ ($i \ge 1$) is nilpotent mod $pA$.

**Lemma** Let $\mathcal{H}$ be the ideal of $A[y]$ generated by $h_i$ ($i \ge 1$). We define the family $(Q_n)_{n \in \mathbb{N}}$ of $A[y]$ by $Q_0(y) = 0$ and $Q_{n+1}(y) = G(y - H(Q_n(y)))$ for $n \in \mathbb{N}$. Then:

1) $(\forall n \in \mathbb{N}) Q_n(y + H(G(y))) = G(y) \mod \mathcal{H}^n$,

2) $(\forall n \in \mathbb{N}) Q_{n+1}(y) = Q_n(y) \mod \mathcal{H}^n$.

**Proof of the lemma** We prove 1) and 2) by induction. For $n = 0$ it is trivial$^3$. Let $n \in \mathbb{N} \setminus \{0\}$, suppose 1) and 2) hold at the step $n - 1$, then modulo $\mathcal{H}^n$:

$Q_n(y + H(G(y))) = G(y + H(G(y)) - H(Q_{n-1}(y + H(G(y))))) = G(y)$ and $Q_{n+1}(y) = G(y - H(Q_n(y))) = G(y - H(Q_{n-1}(y))) = Q_n(y)$.

We return to the proof of Theorem 4. Let $l$ be such that $\mathcal{H}^l \subset pA$ and let $Q(y) = Q_l(y)$. If $F = y+H(px+G(y))$ (resp. $F_1 = y-H(px+Q(y))$) then by 1) (resp. 2)) of the lemma $Q(F) = G(y) \mod pA[x,y]$ (resp. $G(F_1) = Q(y) \mod pA[x,y]$) i.e. there exists $K \in A[x,y]$ (resp. $K_1 \in A[x,y]$) such that $Q(F) = G(y) - pK(x,y)$ (resp. $G(F_1) = Q(y) - pK_1(x,y)$). We define $\sigma = (x+K,F)$ and $\sigma_1 = (x+K_1,F_1)$. We have: $\sigma\sigma_1(y) = F_1(x+K,F) = F - H(px + pK + Q(F) = y$ and $\sigma\sigma_1(x) = x + K + K_1(x+K,F)$. Since $p(K + K_1(x+K,F)) = Q(F) - G + Q(F) - G(F_1(x+K,F)) = 0$ we have $K + K_1(x+K,F) = 0$ because $p \in A^\times$, hence $\sigma\sigma_1(x) = x$. A similar computation shows that $\sigma_1\sigma(y) = y$ and $\sigma_1\sigma(x) = x$, hence $\sigma \in GA_2^2(A)$ and $F \in VA_2^2(A)$.

**Remark** If $Q \subset A$ then Theorem 4 follows from Theorem 2. The first part of Theorem 3.10 in [DY] follows from Theorem 4 and also from Theorem 2.

**Remark** The lemma gives an algorithm to compute $Q_l(y)$ and find a $\sigma \in GA_2^2(A)$ such that $\sigma(y) = y + H(px + G(y))$ (i.e. find $\sigma(x)$) but we don't know any general formula giving $Q_l(y)$ in terms of $p, G$ and $H$. However here is an example of such formula:

**Example** Let $p \in A^\times$, and $G(y) = ay^j$, $H(y) = ry^k$ with $j, k \ge 1$, $r \in A$ nilpotent mod $pA$ and $a \in A$. We can compute the sequence defined in the

<footnote>$^3\mathcal{H}^0 = A$.</footnote>
lemma:

$$ (\forall n \in \mathbb{N}) \quad Q_n(y) = ay^j \sum_{i=0}^{n-1} c_{i,j,k} (-ra^k y^{jk-1})^i \quad \mod r^n A[y] $$

where the coefficients $c_{i,j,k}$ are defined by the formula $C(T) = \sum_{i=0}^{\infty} c_{i,j,k} T^i$ where $C(T) \in A[[T]]$ is defined by $C(T) = (1 + TC(T)^k)^j$.

In fact, by induction, modulo $r^{n+1}A[y]$:

$$ \begin{aligned} Q_{n+1}(y) &= G(y - H(Q_n(y))) \\ &= a \left( y - r \left[ ay^j \sum_{i=0}^{n-1} c_{i,j,k} (-ra^k y^{jk-1})^i \right]^k \right)^j \\ &= ay^j \left( 1 - ra^k y^{jk-1} \left[ \sum_{i=0}^{n-1} c_{i,j,k} (-ra^k y^{jk-1})^i \right]^k \right)^j \\ &= ay^j \sum_{i=0}^{n} c_{i,j,k} (-ra^k y^{jk-1})^i. \end{aligned} $$

So we can take one<fnref n="4" /> of the following automorphisms:

$$ \left\{ \begin{aligned} \sigma(x) &= x + p^{-1}a \left( y^j - \sigma(y)^j \sum_{i=0}^{l-1} c_{i,j,k} (-ra^k \sigma(y)^{jk-1})^i \right) \\ \sigma(y) &= y + r(px + ay^j)^k. \end{aligned} \right. $$

**Example** With $A = k[z, t]$, $p = z$, $G(y) = ty$ and $H(y) = zy^2$, we have the following automorphism $\sigma$ of $k[z, t][x, y]$:

$$ \left\{ \begin{aligned} \sigma(x) &= x - t(zx + ty)^2 \\ \sigma(y) &= y + z(zx + ty)^2. \end{aligned} \right. $$

This is the second Popov's automorphism [P], a variant of which is used by van den Essen and Hubbers in [EH] as counterexample for several conjectures.

**Example** With $A = k[z]$, $p = z^2$, $G(y) = y^2$ and $H(y) = zy^2$, we have the following automorphism $\sigma$ of $k[z][x, y]$:

$$ \left\{ \begin{aligned} \sigma(x) &= x + z^{-2}(y^2 - \sigma(y)^2 + 2z\sigma(y)^5) \\ \sigma(y) &= y + z(z^2x + y^2)^2. \end{aligned} \right. $$

**Questions** One can ask the following questions (the answers should depend on $A$):

1. Let $U(A) = \{(x, ay); a \text{ a unit in } A\}$.

Do we have $VA_2^2(A) \setminus VA_2^1(A) \subset U(A)LV_2^2(A)$?

2. Does there exist an integer $n$ such that $GA_2(A)$ is generated by $Af_2(A)$, $GA_2^n(A)$ and $N_2(A)$?

**Remark** The lemma of Theorem 5 is a partial answer to 1.

<footnote n="4">⁴if $b \in A$ and $b = 0 \mod pA$ then $p^{-1}b$ mean one of $c \in A$ such that $b = pc$.</footnote>
# 4 Tameness properties.

Corollary 1 gives:

**Proposition 3** If $A$ is a integral domain, let $F = y + H(px + G(y)) \in LV_2^2(A)$ then $F \in TV_2(A)$ if and only if the three following conditions are satisfied:

1) $g_i = 0 \bmod pA$ for all $i \ge 2$,

2) $\exists s \in A$ such that $pA + g_1A = sA$,

3) $sk_i = 0 \bmod pA$ for all $i \ge 2$ where $H(sy + g_0) = \sum k_iy^i$.

**Remark** The “moreover” part of Theorem 3.10 in [DY] follows from Proposition 3.

**Theorem 5** If $A$ is a UFD, the following inclusion holds $LV_2^2(A) \subset SV_2(A)$.

**Lemma** Suppose $A$ is a UFD. Let $p \in A^\times$, $G(y) = \sum g_iy^i \in A[y]$ and let $H(y) = \sum h_iy^i \in A[y]$. If $\gcd(p, g_i; i \ge 1)$ is a unit and $y + H(px + G(y)) \in VA_2^2(A)$ then for all $i \ge 1$, $h_i$ is nilpotent mod $pA$.

**Proof of the lemma** We write $p = \prod p_j^{n_j}$ with $p_j$ irreducible. For all $j$, $y+\bar{H}(\bar{G}(y))$ is a variable in $(A/p_jA)[y]$. Since $\deg \bar{G} \ge 1$, we have $\deg \bar{H} \le 1$ i.e. for all $i \ge 1$ $h_i = 0 \bmod p_jA$. Hence for all $i \ge 1$ we have $h_i = 0 \bmod \prod p_jA$ which implies $h_i$ is nilpotent mod $pA$.

**Proof of Theorem 5** (this argument follows an idea due to D. Wright$^5$) Let $F = y + H(px + G(y)) \in LV_2^2(A)$, we prove that $F$ is stably tame by induction on the number of irreducible factors of $p$. If $p$ is a unit then $F$ is tame. Suppose $p$ is not a unit. Let $s = \gcd\{p, g_i; i \ge 1\}$.

1) If $s$ is not a unit then we change $p$ to $s^{-1}p$, $G(y)$ to $s^{-1}(G(y) - g_0)$ and $H(y)$ to $H(sy + g_0)$. This does not change $F$ and the number of irreducible factors of $p$ decreases.

2) If $s$ is a unit, by the lemma $r = \gcd\{p, h_i; i \ge 1\}$ is not a unit. Let $\sigma \in GA_2^2(A)$ such that $\sigma(y) = F$ and let $z = x_3$ be a new indeterminate, we extend $\sigma$ to $\sigma_1$ by $\sigma_1(x) = \sigma(x)$, $\sigma_1(y) = \sigma(y)$ and $\sigma_1(z) = z$. We write $H(y) = rH_1(y)$ (one can suppose $h_0 = 0$) with $H_1 \in A[y]$. We consider$^6$ $\alpha = (x, z, y + rz)$, $\beta = (z, y, x)$, $\gamma = (y, -ry + z, x) \in Af_3(A)$ and $\tau = (x - H_1(G(y) + pz), y, z) \in BA_3(A)$. Let $\sigma_2 = \gamma\tau\beta\sigma_1\alpha$, we have $\sigma_2(z) = z$ and $\sigma_2(y) = y - H_1(px + G(-ry + z))$. Let $G^*(y) = \sum g_i^*y^i = G(w - ry) \in A[w][y]$, now $r$ divides $s^* = \gcd\{p, g_i^*; i \ge 1\}$ and we can reduce the number of irreducible factors of $p$ as in case 1).

<footnote>$^5$Private communication.</footnote>

<footnote>$^6$$\rho = (f, g, h)$ mean $\rho(x) = f$, $\rho(y) = g$, $\rho(z) = h$.</footnote>
**Proposition 4** Let $F \in VA_2^1(A)$ then there exists $\alpha \in Af_2(A)$ such that
$\alpha(F) \in LV_2^2(A)$.

**Proof** Let $F = px + G(y) \in VA_2^1(A)$ with $G(y) = \sum g_i y^i$. Let $a, b \in A$ be such that $ag_1 - pb = 1$. We consider $\alpha = (g_1x - by, -px + ay) \in Af_2(A)$, then $\alpha(F) \in LV_2^2(A)$.

**Corollary 3** If A is a UFD, the following inclusion holds $VA_2^1(A) \subset ST_2(A)$.

**Example** Let $\epsilon \in \mathbb{C}$ be such that $\epsilon^2 + 3 = 0$. By Theorem 3, $2(1 + \epsilon)x + y + 2y^2 + (1 + \epsilon)y^3 \in VA_2^1(\mathbb{Z}[\epsilon])$, the proof of Theorem 5 can not be applied to this variable because the terms $2y^2$ and $(1 + \epsilon)y^3$ can not be canceled simultaneously. So we don't know whether this variable is stably tame or not.

**Remark** Corollary 3 implies that Nagata polynomial is stably tame which is a result due to M. Smith [S].

**Definition** If $\mathbb{Q} \subset A$, we say that $\sigma \in GA_2(A)$ is a Smith automorphism if there exists a triangular derivation $D$ of $A[x, y]$ and $W \in \ker D$ such that $\sigma = \exp(WD)$; we say that $F \in A[x, y]$ is a Smith variable if there exists a Smith automorphism $\sigma$ such that $\sigma(y) = F$.

**Remark** M. Smith's result [S] asserts that a Smith automorphism is stably tame.

**Theorem 6** If A is a UFD and $\mathbb{Q} \subset A$, let $F = y + H(px + G(y)) \in LV_2^2(A)$ and suppose $H \notin A$, then $F$ is a Smith variable if and only if $H(G(y))G'(y) = 0 \mod pA[y]$.

**Proof** We write $p = \prod_{i \in I} p_i^{l_i}$ ($p_i$ irreducible in $A$) and for $i \in I$ we denote by $v_i$ the $p_i$-adic valuation in $A[y]$. We have the following equivalence ($K \in A[y]$): $K = 0 \mod pA[y] \Leftrightarrow (\forall i \in I) v_i(K) \ge l_i(*)$.

1) Suppose that $y + H(px + G(y))$ is a Smith variable then there exists a triangular derivation $D$ of $A[x, y]$ and $W \in \ker D$ such that $\exp(WD)(y) = y + H(px + G(y))$. We write $D = B(y)D_x + bD_y$ with $B \in A[y]$ and $b \in A$ and we obtain $bW = H(px + G(y))$. We have $v_i(b) \le v_i(H(px + G(y)))$ for $i \in I$ and $W = b^{-1}H(px + G(y))$. Hence $0 = D(W) = B(y)b^{-1}pH'(px + G(y)) + G'(y)H'(px + G(y))$ and $B(y)p + bG'(y) = 0$ because $H \notin A$. Finally $v_i(p) \le v_i(b) + v_i(G'(y)) \le v_i(px + H(G(y))G'(y))$ for $i \in I$ and we obtain $H(G(y))G'(y) = H(px + G(y))G'(y) = 0 \mod pA[y]$ using (*).

2) Let $h = \gcd(H(G(y)), p)$, suppose that $H(G(y))G'(y) = 0 \mod pA[y]$
then $-p^{-1}hG'(y) \in A[y]$. Let $D = -p^{-1}hG'(y)D_x + hD_y$ this is a triangular derivation of $A[x, y]$, and let $W = h^{-1}H(px + G(y))$. We have $D(W) = -G'(y)H'(px + G(y)) + G'(y)H'(px + G(y)) = 0$, i.e. $W \in \ker(D)$. We have $(WD)(y) = H(px + G(y))$ and $(WD)^j(y) = 0$ for $j \ge 2$, hence $\exp(WD)(y) = y + H(px + G(y))$ and $y + H(px + G(y))$ is Smith variable.

**Example** If $k$ is a UFD and $\mathbb{Q} \subset k$ then, $y+z(zx+ty)^2$ is a Smith variable in $A = k[z, t]$ and $y+z(z^2x+y^2)^2$ is not a Smith variable in $A = k[z]$.

**Proposition 5** If $A$ is a UFD and $\mathbb{Q} \subset A$ then all Smith variables are variables are in $LV_2^2(A)$.

**Proof** Let $D$ be a triangular derivation of $A[x, y]$ and $W \in \ker D$. We write $D = B(y)D_x + bD_y$ with $B \in A[y]$ and $b \in A$. We choose$^7$ $p = b$, $G(y) = \int_0^1 B(y)$ and we remark that the polynomial $bW(b^{-1}x-b^{-1}\int_0^1 B(y), y)$ is in $A[x]$ because $D_y(bW(b^{-1}x-b^{-1}\int_0^1 B(y), y)) = 0$ since $DW = 0$, hence we can choose $H(x) = bW(b^{-1}x-b^{-1}\int_0^1 B(y), y)$. Now we have $\exp(DW)(y) = y + H(px + G(y)) \in LV_2^2(A)$.

**Remark** Proposition 5 shows that Theorem 5 generalizes M. Smith's criterion.

# 5 Transfer.

Theorem 3 and Theorem 4 can be used to find variables of $VA_n(A)$ ($n \ge 2$) by induction on $n$.

**Notations** Let $Y$ be a non empty set of indeterminates, $y$ be an element of $Y$ and $Z = Y \setminus \{y\}$.

**Theorem 7** Let $p \in A^\times$, let $a$ be a unit mod $pA$, and let $G, F \in A[Y]$. If all the coefficients of $G$ are nilpotent mod $pA$ and if $F$ is a variable in $A[Y]$ then $aF(Y) + G(Y) + px$ is a variable in $A[x, Y]$.

**Proof** Let $\alpha \in \mathrm{Aut}_A A[Y]$ such that $\alpha(y) = F(Y)$. By Theorem 3, there exists $\sigma \in \mathrm{Aut}_{A[Z]} A[Z][x, y]$ such that $\sigma(y) = ay + \alpha^{-1}G(Y) + px$. We extend $\alpha$ and $\sigma$ to automorphisms of $A[x, Y]$ by $\alpha(x) = x$ and $\sigma(z) = z$ for all $z \in Z$. Then: $\alpha\sigma(y) = \alpha(ay + \alpha^{-1}G(Y) + px) = aF(Y) + G(Y) + px$.

<footnote>$^7\int_0^1 B(y)$ is the unique polynomial $C \in A[y]$ such that $C'(y) = B(y)$ and $C(0) = 0$.</footnote>
**Corollary 4 (generalised Choudary-Dimca's hypersurfaces)**

Let $m \in \mathbb{N} \setminus \{0\}$. Let $p_i \in A[z_i]$ and $G_i \in A[x_0, \dots, x_{i-1}, z_1, \dots, z_i]$ for $1 \le i \le m$. Let $r_i \in A[z_i]$ be nilpotent mod $p_i A[z_i]$ ($1 \le i \le m$). Then $x_0 + \sum_{i=1}^m (r_i G_i + x_i p_i)$ is a variable in $A[x_0, \dots, x_m, z_1, \dots, z_m]$.

**Proof** By induction on $m$. Theorem 3 applied to the ring $A[z_1]$ implies that $x_0 + r_1 G_1 + p_1 x_1$ is a variable in $A[x_0, x_1, z_1]$ ($x_1 = x, x_0 = y$).

If $x_0 + \sum_{i=1}^m (r_i G_i + x_i p_i)$ is a variable in $A[x_0, \dots, x_m, z_1, \dots, z_m]$ then Theorem 7 applied to the ring $A[z_{m+1}]$ with $Y = \{x_0, \dots, x_m, z_1, \dots, z_m\}$ implies that $x_0 + \sum_{i=1}^{m+1} (r_i G_i + x_i p_i)$ is a variable in $A[x_0, \dots, x_{m+1}, z_1, \dots, z_{m+1}]$.

**Remark** In the case $A = \mathbb{C}$, $p_i = z_i^{d_i-1}$, $G_i = x_{i-1}^{d_i-1}$ and $r_i = z_i$ for $1 \le i \le m$, these are Choudary-Dimca's hypersurfaces (cf. [CD]) and this corollary gives an answer to question 1 in [CD].

**Theorem 8** Let $p \in A^\times$, let $H(y) = \sum h_i y^i \in A[y]$, and let $G, F \in A[Y]$. If $h_i$ ($i \ge 1$) is nilpotent mod $pA$ and if $F$ is a variable in $A[Y]$ then $F(Y) + H(px + G(Y))$ is a variable in $A[x, Y]$.

**Proof** Let $\alpha \in \mathrm{Aut}_A A[Y]$ such that $\alpha(y) = F(Y)$. By Theorem 4, there exists $\sigma \in \mathrm{Aut}_{A[Z]} A[Z][x, y]$ such that $\sigma(y) = y + H(px + \alpha^{-1}G(Y))$. We extend $\alpha$ and $\sigma$ to automorphisms of $A[x, Y]$ by $\alpha(x) = x$ and $\sigma(z) = z$ for all $z \in Z$. Then: $\alpha\sigma(y) = \alpha(y + H(px + \alpha^{-1}G(Y))) = F(Y) + H(px + G(Y))$.

**Examples** Using Theorem 7 and Theorem 8 one can prove that the following polynomials of $A[x, y, z, u, v]$ are variables:

$$ (u-1)(y+z(z^2x+yz+y^3)^4+u(u+1)x^2+u^3(u+1)^2v, \\ (z-1)y+z(z+1)y^2+z^3(z+1)^2x+u(u^3v+y^3)^4 \text{ and} \\ y+z(z^2x+y^2)^2+u(uv+xyz)^3. $$

**Acknowledgements** We are grateful to the organizers of the conference Poly'99 for their warm hospitality. We'd like to thank J. T. Yu who sent us a copy of [DY] and D. Wright who helped us to elaborate the proof of Theorem 5.

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