# Stratification of projection maps from toric varieties

Boulos El Hilany, Martin Helmer, Elias Tsigaridas

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Boulos El Hilany, Martin Helmer, Elias Tsigaridas. Stratification of projection maps from toric varieties. Journal of Symbolic Computation, 2026, 135, pp.102547. 10.1016/j.jsc.2025.102547. hal-05440865

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# Stratification of projection maps from toric varieties

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Boulos El Hilany<sup>a</sup>, Martin Helmer<sup>b</sup>, Elias Tsigaridas<sup>c</sup>

## ARTICLE INFO

**Article history:**

Received 13 November 2025

Accepted 19 December 2025

Available online 29 December 2025

**Keywords:**

Whitney stratification

Toric geometry

Binomial ideals

Projections of varieties

Polynomial maps

## ABSTRACT

We define a polyhedral version of a stratification for projection maps that applies to any complex or real toric variety and show that it yields similarly desirable properties to the classical map stratification of a proper map. Our results are constructive and give rise to a method for associating the Whitney strata of the projection to the faces of the polytope of the corresponding toric variety. For all the examples we consider, our resulting algorithm outperforms known general purpose methods, e.g., Helmer and Nanda (FoCM, 2022), and Đinh and Jelonek (DCG, 2021), for computing map stratifications.

© 2025 The Author(s). Published by Elsevier Ltd. This is an open

access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

## 1. Introduction

A Whitney stratification of a complex variety is its decomposition into simpler components called strata, which are smooth manifolds that fit together in a prescribed (and desirable) way. One can also expand this notion to include analytic maps, which gives rise to a piece-wise fibration. This procedure was first introduced to study singularities of analytic maps and to classify manifolds Golubitsky and Guillemin (1973); Whitney (1965); Levine (2006). We can also extend this decomposition to apply

E-mail addresses: b.el-hilany@tu-braunschweig.de, boulos.hilani@gmail.com (B. El Hilany), martin.helmer@swansea.ac.uk (M. Helmer), elias.tsigaridas@inria.fr (E. Tsigaridas).

https://doi.org/10.1016/j.jsc.2025.102547

0747-7171/© 2025 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

<footnote><sup>a</sup> Institut für Analysis und Algebra, TU Braunschweig, Universitätsplatz 2, 38106 Braunschweig, Germany</footnote>

<footnote><sup>b</sup> Room 337, The Computational Foundry, Swansea University, Bay Campus, Fabian Way, Swansea, SA1 8EN, Wales, UK</footnote>

<footnote><sup>c</sup> Inria Paris and Institut de Mathématiques de Jussieu - Paris Rive Gauche, Equipe Combinatoire et Optimisation, Sorbonne Université - Campus Pierre et Marie Curie, Case courrier 247, 4, place Jussieu, 75252 Paris Cedex 05, France</footnote>
to real (semi-)algebraic sets, where the corresponding algorithms are mainly dependent on (variants of) cylindrical algebraic decomposition Mather (2012); in this case the applications include, among others, mathematical physics, PDEs, optimization, and data science, e.g., Helmer and Nanda (2022a); Pflaum (2001); Đinh and Jelonek (2021); Dickenstein et al. (2019).

We can express the Whitney stratification of a variety, or a map, using the language of the first order theory of the reals. This results in algorithms for computing the stratification based on quantifier elimination techniques. Their worst-case complexity bound is singly exponential with respect to the number of variables (see, e.g., Mostowski and Rannou (1991); Rannou (1998)), which would be an optimal scenario for implementations. However, to the best of our knowledge, such implementations do not exist. This is so, mainly due to the nature of quantifier elimination algorithms because of their dependence on infinitesimals for computations. In turn, it contributes to a general lack of practicality of such algorithms. On the other hand, in practice, all computer implementations of real quantifier elimination rely on cylindrical algebraic decomposition, which, not only has doubly exponential worst case complexity (with respect to the number of variables), but also tends to exhibit this worst case behavior on the majority of examples.

Using a different approach, Đinh and Jelonek (2021) proposed a technique to compute the Whitney stratification of a polynomial map over an affine algebraic variety using asymptotic critical values of polynomial maps. Their methods are based on Gröbner basis computations from elimination theory. However, the resulting algorithm becomes almost unusable in practice, as the number of variables needed in intermediate computations is more than four times the number of variables appearing in the input. Rather recently, Helmer and Nanda Helmer and Nanda (2022b,a, 2023a) overcame this obstacle by relying on the relation of Whitney stratification with the conormal varieties Lê Tráng and Teissier (1988); Flores and Teissier (2018). Their algorithms only doubles the number of variables and even though it has double exponential worst case (due to Gröbner basis computations) it is efficient in practice; we refer the interested reader to Helmer and Nanda (2022b) for further details and experimental results. We should also point out that some of the above results were very recently extended to real algebraic varieties Helmer and Nanda (2023b). We notice that for a restricted class of real varieties, Vorobjov Vorobjov (1992) suggested a specialized algorithm for Whitney stratification based on quantifier elimination, that suffers from the same (practical) drawbacks as the similar algorithms for complex varieties.

The algorithms mentioned above apply to all complex and real algebraic varieties and can be extended to polynomial maps over them. Furthermore, no restriction or assumption on the input is required. However, to mention the main drawback, one is forced to disregard any symmetry, combinatorics, or geometry of the objects at hand. The latter structure is an important characteristic of varieties that emanate from applications. Consequently, a reasonable approach to accelerate the computationally intensive Whitney stratification algorithms is to study the relationship between strata and structure.

We consider the problem of computing a Whitney stratification of projection maps on toric affine varieties. Accordingly, we obtain a stratification of the parameter space of any family of affine varieties, whose ideal is generated by binomial equations. Thus, each stratum in the range corresponds to a subset of diffeomorphic binomial varieties. Such varieties appear frequently, for example, in chemistry as mathematical models that describe chemical reaction networks Adamer and Helmer (2019); Pérez Millán et al. (2012).

A Whitney stratification method for a (real and complex) toric variety $X_A$ corresponding to any finite set $A \subset \mathbb{Z}^v$ of lattice points already exists in (Gelfand et al., 1994, Theorem 5.3. §11.5.B). Namely, each stratum coincides with a $k$-orbit (with $k = \mathbb{R}_{\ge 0}$ or $k = \mathbb{C}$) of $X_A$. In turn, the faces of $A$ provide an accurate description of the strata of $X_A$. Consequently, any projection map $\pi : X_A \to k^m$, which forgets some coordinates, is expected to have a similar stratified description.

We consider any pair $(X_A, \pi)$ as above and we describe a subdivision of the space $k^m$, for which the projection restricted to each stratum has diffeomorphic fibers.

Altogether, the strata are in bijection with the set of faces of $\text{conv}(A)$. Our work gives rise to an algorithm that takes $X_A$ as an input (using either $A$ or the corresponding binomial representation) and produces a subdivision of $X_A$ and of $k^m$. The subdivision of $X_A$ arises from set differences of toric varieties $X_B$, for some subset $B \subset A$ that lies in a face of the polytope $\text{conv}(A)$. The subdivision of
$k^m$ arises from the set difference of coordinate subspaces of $k^m$ (of different dimensions) determined by the faces of conv($\mathcal{A}$).

Our main result, together with the necessary notations, is presented in §2. In §3 we present the proof of our main Theorem 3; we show how to use the structure of toric ideals to construct diffeomorphisms between fibers of the projection over the same stratum.

Theorem 3 implies that the face-lattice of conv($\mathcal{A}$) plays an important role in computing the strata of the projection map. The running time of our resulting Algorithm 1 is dominated by (various) combinatorial tasks. Consequently, this provides a significant (overall) run-time speedup. We provide some experimental evidence of this by producing several examples in §4 and making a comparison in Table 1. In §4.2 we present a detailed Example to illustrate Theorem 3 and the functionality of Algorithm 1.

### Acknowledgments

We are grateful to the referee of an earlier version of our paper for her/his very constructive comments that helped to us to improve a lot the presentation of our results. Martin Helmer is supported by the Royal Society under grant RSWF/R2/242006 and by the Air Force Office of Scientific Research (AFOSR) under award number FA9550-22-1-0462, managed by Dr. Frederick Leve, and would like to gratefully acknowledge this support. Elias Tsigaridas is partially supported by the PGMO grant SOAP and ANR PRC ZADyG (ANR-25-CE48-7058).

## 2. Statement of the main theorem

We begin by briefly reviewing the required notations and definitions.

Let $Y$ be an algebraic variety of dimension $d$. We say a filtration $Y_\bullet$ of varieties $Y_0 \subset \dots \subset Y_d = Y$ is a *Whitney stratification of $Y$* if, for all $i$, $Y_i$ is a variety of dimension $i$ and $Y_i - Y_{i-1}$ is a smooth manifold such that *Whitney's condition B* holds for all pairs $M, N$, where $M$ is a connected component of $Y_i - Y_{i-1}$ and $N$ is a connected component of $Y_j - Y_{j-1}$. Such connected components are called *strata*. A pair of strata, $M, N$ with $\bar{M} \subset \bar{N}$, satisfy Whitney's condition B at a point $x \in M$ with respect to $N$ if: for every sequence $\{p_n\} \subset M$ and every sequence $\{q_n\} \subset N$, with $\lim_{p_n = \lim_{q_n = x}} s \subset T$ where $s$ is the limit of secant lines between $p_n, q_n$ and $T$ is the limit of tangent plans to $N$ at $q_n$. We say the pair $M, N$ satisfies condition B if condition B holds, with respect to $N$, at all points $x \in M$.

A stratification of a polynomial map of algebraic varieties is defined as follows.

**Definition 1.** Let $Y_\bullet$ and $Z_\bullet$ be Whitney stratifications of algebraic varieties $Y$ and $Z$. The pair $(Y_\bullet, Z_\bullet)$ is a **fibrated stratification** of a polynomial map $\phi : Y \to Z$ if for each stratum $R \subset Y$ there exists a stratum $S \subset Z$ such that:

(1) the image $\phi(R)$ is wholly contained in $S$,

(2) the restricted map $\phi|_R : R \to S$ is a smooth submersion, i.e., its differential $d(\phi|_R)_x : T_xR \to T_{\phi(x)}S$ is surjective at each point $x$ in $R$, and

(3) for each stratum $S \subset Z$, the two fibers $\phi|_R^{-1}(p)$ and $\phi|_R^{-1}(q)$ are diffeomorphic for any two points $p, q \in S$.

A fibrated stratification satisfies the following further assertion that is relevant to the applications in chemical reaction network theory mentioned above.

**Lemma 2.** Let $\phi : Y \to Z$ be the map satisfying the three assertions of Definition 1. Then, for any stratum $S \in Z_\bullet$, and any two points $p, q \in S$, the two fibers $\phi^{-1}(p)$ and $\phi^{-1}(q)$ are diffeomorphic.

**Proof.** Let $R_1, \dots, R_\ell \in Y_\bullet$ denote the collection of strata of $Y$ mapping to $S$. Item 3 shows that there is a family of diffeomorphisms $\{\Psi_i : \phi|_{R_i}^{-1}(p) \to \phi|_{R_i}^{-1}(q)\}_{i=1,\dots,\ell}$. Consider the map
$$ \Psi : \phi|_{R_1}^{-1}(p) \cup \dots \cup \phi|_{R_l}^{-1}(p) \to \phi|_{R_1}^{-1}(q) \cup \dots \cup \phi|_{R_l}^{-1}(q), $$

given as $\Psi(y) := \Psi_i(y)$ whenever $y \in R_i$ for some $i = 1, \dots, l$. This map is well-defined since strata $R_i$ are pairwise disjoint. Since differentiability of a map between (possibly non-connected) manifolds is a local question and is independent of the particular local representation used, the map $\Psi$ is itself a diffeomorphism. Furthermore, from the definitions above, we have $R_1 \cup \dots \cup R_l = \phi^{-1}(S)$. Therefore, the domain and range of $\Psi$ are $\phi^{-1}(p)$ and $\phi^{-1}(q)$ respectively. The map $\Psi$ thus constructed forms the required diffeomorphism in Lemma 2. $\square$

Consider a polynomial map $\phi : Y \to Z$ between varieties $Y$ and $Z$. Thom's First Isotopy Lemma (Mather, 2012, Proposition 11.1) shows that whenever $\phi$ is proper, if the stratification $(Y_\bullet, Z_\bullet)$, with both $Y_\bullet$ and $Z_\bullet$ being Whitney, satisfies Assertions (1) and (2) of Definition 1, then it satisfies Assertion (3). We note that, in the general case of polynomial maps between algebraic varieties, none of the conditions above (i.e., that the map is proper, that the domain and codomain are Whitney stratified, and that for each strata the conditions in items (1) and (2) hold) can be relaxed to guarantee (3), however for many particular examples weaker conditions would suffice (see, e.g., Đinh and Jelonek (2021)). While the maps of interest to us may not be proper, we still obtain an analogous result for a certain class of varieties (namely *toric varieties*); below we recall some important definitions.

To any monoid $(\mathcal{M}, +)$, we can associate the set $\mathcal{G}(\mathcal{M})$ so that $(\mathcal{G}(\mathcal{M}), +)$ is the smallest group containing $\mathcal{M}$. Thus, we have $\mathcal{G}(\mathbb{R}_{\ge 0}) = \mathbb{R}$ and $\mathcal{G}(\mathbb{C}) = \mathbb{C}$. We seek to study solutions to systems of parametric binomials in $k^n$ where $k = \mathbb{R}_{\ge 0}$ or $k = \mathbb{C}$. When $k = \mathbb{R}_{\ge 0}$ we hence seek to study polynomials with coefficients in $\mathcal{G}(\mathbb{R}_{\ge 0}) = \mathbb{R}$ while when $k = \mathbb{C}$ we will want coefficients in $\mathcal{G}(\mathbb{C}) = \mathbb{C}$. To avoid having to write the unwieldily expression $\mathcal{G}(k)$ we define $\tilde{k} := \mathcal{G}(k)$.

With this notation in hand, let $k \in \{\mathbb{R}_{\ge 0}, \mathbb{C}\}$, and consider a parametric system of binomial equations which define a prime ideal. That is, work in a ring $\tilde{k}[x, c]$ where $x = x_1, \dots, x_n$ are thought of as variables and $c = c_1, \dots, c_m$ are thought of as parameters. Let $\mathcal{A} = \{a(1), \dots, a(n+m)\} \subset \mathbb{Z}^v$ be a finite subset having $n+m$ elements. Let $I_{\mathcal{A}} = \langle f_1, \dots, f_r \rangle$ be a prime binomial ideal in $\tilde{k}[x, c]$ such that

$$ Y = X_{\mathcal{A}} = \mathbb{V}(I_{\mathcal{A}}) = \overline{\{ (t^{a(1)}, \dots, t^{a(n+m)}) \mid t \in (k^*)^v \}}. \quad (2.1) $$

The closure in (2.1) refers to the Zariski closure in the affine variety $k^{n+m}$. Then, for each parameter choice $p \in k^m$, we obtain a variety

$$ Y(p) = \mathbb{V}(f_1(x, p), \dots, f_r(x, p)) \subset k^n. $$

If $Y$ is given as in (2.1), it will be called an *affine toric variety*, whereas the variety $Y(p)$ above will be referred to as *affine binomial* – the latter type specializes to the former for each choice of $p$.

Our goal is to study toric varieties given by the parametric systems of binomial equations above and to obtain a stratification of the parameter space $Z = k^m$, such that for each stratum $S$ of $Z$, and for any $p, q \in S$, the two binomial varieties $Y(p)$ and $Y(q)$ are homeomorphic.

Let $\pi : Y \to k^m$ be the coordinate projection $(x, c) \mapsto c$. Then, thanks to (2.1), we have that $\overline{\pi(Y)} = X_B$, where $B = \{a(n+1), \dots, a(n+m)\} \subset \mathbb{Z}^v$ is the set containing the last $m$ elements of $\mathcal{A}$.

Let $J \subset [n+m]$, and let $T_J$ denote the coordinate subspace of $k^{n+m}$ given by:

$$ T_J := \{y := (x, c) \in k^{n+m} \mid j \in J \implies y_j = 0\}. $$

We use the convention $T_\emptyset = k^{n+m}$.

For any finite subset $S := \{s(1), \dots, s(\ell)\} \subset \mathbb{Z}^v$, and for each face $\Gamma$ of $\mathrm{conv}(S)$, we define $J_\Gamma \subset [\ell]$ such that

$$ j \in J_\Gamma \iff s(j) \in S - \Gamma. \quad (2.2) $$

In this notation $T_{J_\Gamma}$, for a face $\Gamma$, has the effect of setting to zero all monomials whose exponent vectors do not belong to $\Gamma$. We now state our main result.
**Theorem 3.** Let $Y := X_A$ be the affine toric variety in $k^{n+m}$ defined by a prime binomial ideal $I_A = \langle f_1, \dots, f_r \rangle$ in $\bar{k}[x, c]$, treating $x$ as variables and $c$ as parameters, and let $\pi : Y \to k^m$ be the coordinate projection $(x, c) \mapsto c$. Then, we have $Z := \pi(Y) = X_B$, where $B = \{a(n+1), \dots, a(n+m)\} \subset \mathbb{Z}^\nu$ is the set containing the last $m$ elements of $A$. Use the notations above and consider the decomposition of the toric variety $Y$ as $Y_\bullet := Y_0 \subset \dots \subset Y_d$, where

$$Y_i := \bigcup_{\substack{\Gamma \text{ face of } \mathrm{conv}(A) \\ \mathrm{s.t. } \dim(\Gamma)=i}} Y \cap T_{J_\Gamma},$$

and the decomposition of the toric variety $Z$, as $Z_\bullet := Z_0 \subset \dots \subset Z_d$, where

$$Z_i := \bigcup_{\substack{\Gamma \text{ face of } \mathrm{conv}(B) \\ \mathrm{s.t. } \dim(\Gamma)=i}} Z \cap T_{J_\Gamma}.$$

Then, the pair $(Y_\bullet, Z_\bullet)$ is a fibrated stratification (see Definition 1) of the map $\pi$.

**3. Proof of the main result and associated lemmas**

This section is devoted to proving Theorem 3. In what follows, we retain all notations from §2. To prove Theorem 3 we need to show several properties (as in Definition 1) hold for restrictions of the map $\pi$ to each (open, connected) stratum of the domain. By Lemma 6 below we have that all open and connected strata arising from $Y_\bullet$ as in Theorem 3 have the form<fnref n="1" />

$$Y \cap T_{J_\Gamma} - Y_{\dim(\Gamma)-1} \cong X_D - \mathbb{V}(z_1 \cdots z_\mu) = X_D \cap (k^*)^\mu, \quad (3.1)$$

for $X_D$ a toric variety in some affine space $k^\mu$ and using coordinates $z_i$ on $k^\mu$.

Let $V \subset k^{n+m}$ be any affine toric variety with corresponding set of lattice points $A \subset \mathbb{Z}^\nu$. We shall also need to define some additional notations. In what follows, we fix a set $A := \{a(1), \dots, a(n+m)\} \subset \mathbb{Z}^\nu$, and consider the following sequence of functions

$$(k^*)^\nu \xrightarrow{\Phi_A} (k^*)^{n+m} \to (k^*)^m, \quad (3.2)$$

where the second map is the projection $(x, c) := (y_1, \dots, y_{n+m}) \mapsto (y_{n+1}, \dots, y_{n+m}) =: c$, and $\Phi_A$ is the toric map $t \mapsto (t^a)_{a \in A}$. We define $V^* := V \cap (k^*)^{n+m}$. Consider the map

$$F := \pi|_{V^*} : V^* \to \pi(V^*). \quad (3.3)$$

As noted above, for our stratification $Y_\bullet$, all (open and connected) strata have the form of Eq. (3.1), and hence all restriction maps appearing in Definition 1 have the form of the map $F$ above. We remark that $F$ can be viewed, equivalently, as either a map of very affine varieties (i.e., subvarieties of the algebraic torus) or as the restriction of a map of affine varieties. For the purposes of the result in Theorem 3 we require the latter perspective, however for the steps of proof, i.e., in Propositions 4 and 5, we will employ the former.

The proof of Theorem 3 is postponed to the end of this section. It requires in particular showing that $F$ satisfies the three properties (containment, submersion, and diffeomorphism) listed in Definition 1, as well as that the subdivisions $Y_\bullet$ and $Z_\bullet$ define Whitney stratifications of $Y$ and $Z$, respectively. We first show that $F$ is a submersion in Proposition 4, while the diffeomorphism property is treated in Proposition 5. The final aspects are given in the proof of Theorem 3 at the end of this section.

<footnote n="1">¹ This form is up to composition with a linear embedding. In particular, if $\rho = n+m$ and $\mu \le \rho$ then there is an implied embedding $\iota : k^\mu \to k^\rho$, $\iota : (z_1, \dots, z_\mu) \mapsto \sigma((z_1, \dots, z_\mu, 0, \dots, 0))$ for a (strata-wise fixed) permutation $\sigma$ on $\rho$ letters. We then have $\iota(X_D \cap (k^*)^\mu) = Y \cap T_{J_\Gamma} - Y_{\dim(\Gamma)-1}$.</footnote>
**Proposition 4.** The map F is a submersion.

**Proof.** It suffices to prove that the Jacobian of the map F is of full rank. Notice that $V^* \subset (k^*)^{n+m}$ and that the ideal $I(V^*)$ is generated by the following set of binomials

$$f_i := y^{\beta(i)} - 1, \quad \text{for } i = 1, \dots, r,$$

for some set of exponents $\{\beta(1), \dots, \beta(r)\} \subset \mathbb{Z}^{n+m}$. Then, the Jacobian of the map F is a $(n+m) \times (m+r)$-matrix $[J_V^T | N]$, where $J_V$ is an $(n+m) \times r$-matrix, and $N$ is a $(n+m) \times m$-matrix; they are as follows

$$J_V := \begin{bmatrix} \partial f_1 / \partial y_1 & \cdots & \partial f_r / \partial y_1 \\ \vdots & \ddots & \vdots \\ \partial f_1 / \partial y_{n+m} & \cdots & \partial f_r / \partial y_{n+m} \end{bmatrix}, \quad \text{and} \quad N := \begin{bmatrix} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 \\ 1 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 1 \end{bmatrix}.$$

Notice that the lower part of $N$ is the identity matrix and hence has rank $m$. Therefore, the (sub)matrix $J_V$, by substituting the binomials $f_i$, becomes

$$J_V = \begin{bmatrix} \beta(1)_{1} y^{\beta(1)}/y_{1} & \cdots & \beta(r)_{1} y^{\beta(r)}/y_{1} \\ \vdots & \ddots & \vdots \\ \beta(1)_{n+m} y^{\beta(1)}/y_{n+m} & \cdots & \beta(r)_{n+m} y^{\beta(r)}/y_{n+m} \end{bmatrix}, \quad (3.4)$$

where $\beta(i) := (\beta(i)_1, \dots, \beta(i)_{n+m})$. Let $B$ be the matrix $(\beta(i)_j)_{j,i}$. One can check that any minor of $J_V$ will be of the form

$$\frac{\prod_{i \in I} y^{\beta(i)}}{\prod_{j \in J} y_j} \cdot M,$$

for some $I \subset [r]$, $J \subset [n+m]$, where $M$ is a minor of $B$. We deduce that the Jacobian $J_V$ has constant rank for all $y \in (k^*)^{n+m}$. This shows that the differential $d(F)_y : T_y V^* \to T_{F(y)} F(V^*)$ is surjective at each point $y \in V^*$. $\square$

Note that we always have that $(1, \dots, 1) \in F(V^*)$.

**Proposition 5.** For every $q \in F(V^*)$, we have $F^{-1}(q)$ is diffeomorphic to $F^{-1}(1, \dots, 1)$.

**Proof.** Assume first that $\nu \ge n+m$. Then, the map $\Phi_{\mathcal{A}}$ from (3.2) is surjective. This implies that $V^* = (k^*)^{n+m}$, and thus the map $F = \pi|_{V^*}$ is simply the coordinate projection $(x, c) \mapsto c$ (see (3.3)). Hence, any two fibers under $F$ are diffeomorphic.

In what follows, we assume that $\nu < n+m$. Let $q \in F(V^*)$. In what follows, we construct a diffeomorphism from $F^{-1}(1, \dots, 1)$ to $F^{-1}(q)$.

First, we have the equality

$$I(F^{-1}(q)) = \{x_1 - t^{a(1)}, \dots, x_n - t^{a(n)}, q_1 - t^{a(n+1)}, \dots, q_m - t^{a(n+m)}\} \cap \tilde{k}[x, c]. \quad (3.5)$$

The ideal intersection operation above eliminates the variables $t$, and, as no variables $c$ appear, we obtain an ideal generated by binomials in $x$. Namely, by applying elimination, we find a subset

$$\{\alpha(1), \beta(1), \dots, \alpha(n+m-\nu), \beta(n+m-\nu)\} \subset \mathbb{N}^n$$

and Laurent monomial functions $\varphi_1, \dots, \varphi_{n+m-\nu} : (k^*)^m \to k^*$, satisfying
$$I(F^{-1}(q)) = \{x^{\alpha(1)} - \varphi_1(q) x^{\beta(1)}, \dots, x^{\alpha(n+m-v)} - \varphi_{n+m-v}(q) x^{\beta(n+m-v)}\}. \quad (3.6)$$

Furthermore, since the coefficients in front of the parameters $t$ in (3.5) have value $-1$, the above monomials functions $\varphi_i$ are monic.

Since at least one point in $V^* \subset (k^*)^{n+m}$ maps to $q$ under $F$, the vanishing locus of $I(F^{-1}(q))$ also contains at least one point $s$. The latter then should satisfy

$$s^{\alpha(i)-\beta(i)} = \varphi_i(q), \quad i = 1, \dots, n+m-v. \qquad (3.7)$$

Consider the map:

$$\Lambda_q : (k^*)^{n+m} \to (k^*)^{n+m} \\ (x_1, \dots, x_n, c_1, \dots, c_m) \mapsto (s_1 \cdot x_1, \dots, s_n \cdot x_n, q_1 \cdot c_1, \dots, q_m \cdot c_m).$$

We now show that the map $\Lambda_q$ furnishes the claimed diffeomorphism between $F^{-1}(1, \dots, 1)$ and $F^{-1}(q)$. Since $\Lambda_q$ is a coordinate-wise scaling, we have $F^{-1}(1)$ is diffeomorphic to $\Lambda_q(F^{-1}(1))$. It is left to check $\Lambda_q(F^{-1}(1)) = F^{-1}(q)$.

To show the inclusion "$\subset$", let $p \in (k^*)^n$ such that $(p, 1, \dots, 1) \in F^{-1}(1, \dots, 1)$; note that all points in $F^{-1}(1, \dots, 1)$ have this form. Applying $\Lambda_q$ to $(p, 1, \dots, 1)$ we obtain $(s_1 \cdot p_1, \dots, s_n \cdot p_n, q_1, \dots, q_m)$, hence we need to show $(s_1 \cdot p_1, \dots, s_n \cdot p_n, q_1, \dots, q_m)$ is in $F^{-1}(q)$. Since, for $i=1, \dots, n+m-v$, the monomial functions $\varphi_i$ above are monic, we get $\varphi_i(1, \dots, 1) = 1$. Hence, it follows from the expression in (3.6) that

$$p^{\alpha(i)-\beta(i)} = 1, \quad i = 1, \dots, n+m-v. \qquad (3.8)$$

Indeed, this follows from the expression in (3.6) as $\varphi_i(1, \dots, 1) = 1$ for $i=1, \dots, n+m-v$. Combining (3.8) with (3.7), we obtain

$$(s \cdot p)^{\alpha(i)} - \varphi_i(q)(s \cdot p)^{\beta(i)} = 0, \quad i = 1, \dots, n+m-v. \qquad (3.9)$$

Thanks to the expression (3.6), we conclude the membership $(s_1 \cdot p_1, \dots, s_n \cdot p_n, q_1, \dots, q_m) \in F^{-1}(q)$.

Concerning "$\supseteq$", we will show that $\Lambda_q^{-1}(F^{-1}(q)) \subset F^{-1}(1)$; this will immediately imply the desired conclusion that $\Lambda_q(F^{-1}(1)) \supset F^{-1}(q)$.

Let $p \in (k^*)^n$ and $q \in (k^*)^n$ such that $(p, q) \in F^{-1}(q)$. Then, thanks to (3.6), we have

$$p^{\alpha(i)-\beta(i)} = \varphi_i(q), \quad i = 1, \dots, n+m-v. \qquad (3.10)$$

Hence, for any $s \in (k^*)^n$ satisfying (3.7), we get

$$\left(\frac{p}{s}\right)^{\alpha(i)} = \left(\frac{p}{s}\right)^{\beta(i)}, \quad i = 1, \dots, n+m-v.$$

$$\text{This yields } \Lambda_q^{-1}(p_1, \dots, p_n, q_1, \dots, q_m) = (p_1/s_1, \dots, p_n/s_n, q_1/q_1, \dots, q_n/q_n) \in F^{-1}(1). \quad \square$$

Before concluding with the proof of the main result we give a small lemma; it follows directly from (Cox et al., 2011, Proposition 2.1.8). Let $J \subset [n+m]$, and let $T_J^*$ denote the sub-torus of $k^{n+m}$ given by:

$$T_J^* := \{y := (x, c) \in k^{n+m} \mid y_j = 0 \Longleftrightarrow j \in J\}.$$

Note that $T_J^* = (k^*)^{n+m}$, and $V^* = Y \cap T_J^*$.

**Lemma 6** (Cox et al., 2011, Proposition 2.1.8). Let $\Gamma$ be any face of $\mathrm{conv}(\mathcal{A})$. Then, we have that

$$Y \cap T_{J_\Gamma}^* \cong \Phi_{\Gamma \cap \mathbb{Z}^v}((k^*)^q),$$

where $q := \dim \Gamma$ and $J_\Gamma$ is defined in (2.2) and $\Phi_{\Gamma \cap \mathbb{Z}^v}$ is the toric map $(k^*)^q \to (k^*)^{\Gamma \cap \mathbb{Z}^v}$.
We finish this section by proving the main theorem.

**Proof of Theorem 3.** Our goal is to show that $(Y_\bullet, Z_\bullet)$ gives a fibrated stratification of the map $\pi$.

Projecting each element in $Y_\bullet$ gives rise to a decomposition $Z_\bullet$ of $\pi(Y)$ that satisfies Property (1) in Definition 1.

As noted at the beginning of this section, by Lemma 6 we have that, up to composition with a linear embedding, all open and connected strata arising from $Y_\bullet$ as in Theorem 3 have the form of Eq. (3.1). Hence, all restriction maps appearing in the properties of Definition 1 have the same form as the map $F = \pi|_{V^*}$, studied in Propositions 4 and 5. Therefore, each restriction map satisfies Property (2) of Definition 1 by Proposition 4 and also Property (3) of Definition 1 by Proposition 5. To show that $(Y_\bullet, Z_\bullet)$ is a fibrated stratification of $\pi$, as in Definition 1, it remains to show that the subdivisions $Y_\bullet$ and $Z_\bullet$ define Whitney stratifications of $Y$ and $Z$, respectively.

Concerning $Y_\bullet$, we know that is a Whitney stratification, since the stratification of a toric variety by the faces of the corresponding polytope is Whitney (see e.g. (Gelfand et al., 1994, Theorem 5.3. §11.5.B)).

Now consider $Z_\bullet$. First note that we obtain a subset $B \subset A \subset \mathbb{Z}^\nu$ by composing the projection $\pi: k^{n+m} \to k^m$, with the monomial map $\Phi: (k^*)^\nu \to k^{n+m}$; in particular this composition is given by the map $(t^a)_{a \in A} \mapsto (t^a)_{a \in B}$. We can describe the resulting map as a monomial map $\Phi_B: (k^*)^\nu \to k^m$, $t \mapsto (t^a)_{a \in B}$. In addition, each stratum $R := R_\Gamma$ in $k^{n+m}$ corresponds to a subset $G \subset A$, where $G := \Gamma \cap A$, for some face $\Gamma$ of $\mathrm{conv}(A)$ (Lemma 6). Thus, the stratum $\pi(R)$ is the image of $(k^*)^\nu$ under the map

$$ \Phi_{B,\Gamma}: (k^*)^\nu \to k^m \\ t \mapsto (\varepsilon_a t^a)_{a \in B}, $$

where for each $a \in B$, the value $\varepsilon_a \in \{0, 1\}$ is

$$ \varepsilon_a := \begin{cases} 1, & \text{if } a \in B \cap G \\ 0, & \text{otherwise.} \end{cases} $$

It follows directly from $B \subset A$ that for each face $\Gamma$ of $\mathrm{conv}(A)$, the set $\Gamma \cap \mathrm{conv}(B)$ is a face of $\mathrm{conv}(B)$. Hence, the set $B \cap G$ is either empty, or is the intersection $B \cap \Gamma'$, for some face $\Gamma'$ of $\mathrm{conv}(B)$. Consequently, $Z_\bullet$ is also stratified by the faces of $\mathrm{conv}(B)$ and the conclusion follows again by (Gelfand et al., 1994, Theorem 5.3. §11.5.B). $\square$

# 4. Algorithm, runtime, and an example

Using the notations defined above, we derive from Theorem 3 the following algorithm, Algorithm 1, to compute a fibrated stratification of the projection map $\pi: Y \to Z$ from the toric variety $Y$ (treating $x$ and $c$ as variables) onto (the closure of) its image $Z = \pi(Y)$ in the parameter space $k^m$ as in Definition 1.

## 4.1. Runtime comparisons

We compare Algorithm 1 to the general purpose algorithms appearing in Helmer and Nanda (2023a); Helmer and Mohr (2024) on several instances. We then conclude this section by presenting Example 4.2 that illustrates the functionality of Algorithm 1. Since the new algorithm, Algorithm 1, follows closely the combinatorial structure of the A-matrix and the simple description of toric varieties, a better performance is expected on examples with large numbers of generators, variables, and higher degrees relative to the general purpose symbolic algorithms which apply to a much broader class of examples.

We remark that we do not include the algorithm of Đinh and Jelonek Đinh and Jelonek (2021) in Table 1 since the algorithm of Đinh and Jelonek (2021) was unable to compute a stratification for the inputs above in under eight hours; in fact, the algorithm of Đinh and Jelonek (2021) was also unable
<table><caption>**Algorithm 1.** Algorithm arising from Theorem 3.</caption><thead><tr><th></th><th>ToricStratMap(π : Y → Z)</th></tr></thead><tbody><tr><td>Input:</td><td>The ideal I<sub>A</sub> = {f<sub>1</sub>, ..., f<sub>r</sub>} defining Y = V(I<sub>A</sub>) and the map π : Y → Z.</td></tr><tr><td>Output:</td><td>A stratification (Y<sub>•</sub>, Z<sub>•</sub>) of π.</td></tr><tr><td>1</td><td>Set Z = π(Y), Y<sub>•</sub> = {Y}, Z<sub>•</sub> = {Z}</td></tr><tr><td>2</td><td>For each f<sub>i</sub> = z<sup>i</sup> ± z<sup>k</sup> set v<sub>i</sub> = l - k</td></tr><tr><td>3</td><td>Set D = {d<sub>1</sub>, ..., d<sub>r</sub>}<sup>T</sup> to be the matrix with rows d<sub>i</sub> and set A = ker D</td></tr><tr><td>4</td><td>Set A to be the set consisting of the columns of A</td></tr><tr><td>4</td><td>Set B ⊂ A to be the subset given by the last m entries of A</td></tr><tr><td>5</td><td>Set P = conv(A)</td></tr><tr><td>6</td><td>For each face Γ of P do:</td></tr><tr><td>7</td><td>    d = dim(Γ)</td></tr><tr><td>8</td><td>    Y<sub>d</sub> = Y<sub>d</sub> ∪ (X<sub>A</sub> ∩ T<sub>J<sub>r</sub></sub>)</td></tr><tr><td>9</td><td>    Γ<sup>′</sup> = conv(Γ ∩ B)</td></tr><tr><td>10</td><td>    e = dim(Γ<sup>′</sup>)</td></tr><tr><td>11</td><td>    Z<sub>e</sub> = Z<sub>e</sub> ∪ (X<sub>B</sub> ∩ T<sub>J<sub>r</sub><sup>′</sup></sub>)</td></tr><tr><td>12</td><td>Return (Y<sub>•</sub>, Z<sub>•</sub>)</td></tr></tbody></table>

<caption>**Table 1**</caption>

Runtimes for the new combinatorial algorithm on several toric map examples. The runtimes of the symbolic alternative algorithms use version 2.11 of the WhitneyStatifications Macualay2 package. All examples are run in Macualay2 version 1.24.05. In the table, - denotes examples which take greater than 8 hours to run. Our implementation of ToricStratMap used in the tests above is available at: http://martin-helmer.com/Software/WSToric.m2.

<table><thead><tr><th>Input</th><th>ToricStratMap</th><th>Symbolic Alg. of Helmer and Nanda (2023a)</th><th>Symbolic Alg. of Helmer and Mohr (2024)</th></tr></thead><tbody><tr><td>π : V(I<sub>1</sub>) → C, see (4.1)</td><td>0.08s</td><td>0.14s</td><td>0.12s</td></tr><tr><td>π : V(I<sub>2</sub>) → C<sup>3</sup>, see (4.2)</td><td>0.7s</td><td>-</td><td>3.0s</td></tr><tr><td>π : V(I<sub>3</sub>) → C<sup>4</sup>, see (4.3)</td><td>3.4s</td><td>-</td><td>5.7s</td></tr><tr><td>π : V(I<sub>4</sub>) → C<sup>4</sup>, see (4.4)</td><td>26.1s</td><td>-</td><td>-</td></tr><tr><td>π : V(I<sub>5</sub>) → C<sup>4</sup>, see (4.5)</td><td>712.5s</td><td>-</td><td>-</td></tr></tbody></table>

to compute a Whitney stratification of the Whitney umbrella, without any maps involved, in 24 hours (see the discussion in Helmer and Nanda (2022a, 2023a)).

In the following, we list the defining equations for the examples appearing in Table 1. As before we consider an ideal $I$ in $\tilde{k}[x_1, ..., x_n, c_1, ..., c_m]$, in our case this is always a binomial ideal, where we consider the $n$-tuple $x$ as variables and the $m$-tuple $c$ as parameters. Now, if we treat both $x$ and $c$ as variables, then ideal correspond to a variety $Y = V(I) \subset k^{n+m}$. Our goal is to stratify the projection map $\pi : Y \to k^m$, $(x, c) \mapsto c$. In the following, we abuse notation by writing the projection in more compact form, i.e., $\pi : V(I) \to k^m$. Note that in all the examples we consider, the map $\pi$ is surjective onto the parameter space, and hence we always have $Z = k^m$.

Our first example, $I_1$, emanates from the Whitney umbrella. Examples $I_3$ and $I_5$ come from chemical reaction network theory; the example $I_3$ comes from the two-site kinetic proofreading model, (Adamer and Helmer, 2019, Example 2.6), while $I_5$ comes from the two-site phosphorylation system as given in (Pérez Millán et al., 2012, Example 3.13). In the cases arising from chemical reaction networks, the $c_i$'s represent (unknown) reaction rate constant parameters, while the $x_i$'s represent chemical concentrations; see, e.g. (Adamer and Helmer, 2019, §2.3), or (Pérez Millán et al., 2012, §2), and the references therein.

$$ \pi : V(I_1) \to \mathbb{C}_c. \quad V(I_1) \subset \mathbb{C}^3, \quad I_1 = \langle c x_2^2 - x_1^2 \rangle. \qquad (4.1) $$

$$ \pi : V(I_2) \to \mathbb{C}_c^3. \quad V(I_2) \subset \mathbb{C}^6, \quad I_2 = \langle c_2 c_3 x_3 - c_1 x_2, c_3 x_1^2 - x_2^2, c_1 x_1^2 - c_2 x_2 x_3 \rangle. \qquad (4.2) $$

$$ \pi : V(I_3) \to \mathbb{C}_c^4. \quad V(I_3) \subset \mathbb{C}^8, \quad I_3 = \langle c_3 x_1 - c_4 x_2, c_1 x_3 x_4 - c_2 x_1 \rangle. \qquad (4.3) $$

$$ \pi : V(I_4) \to \mathbb{C}_c^4. \quad V(I_4) \subset \mathbb{C}^9, \\ I_4 = \langle c_1 x_3 - x_4, x_1^2 x_3 - x_2^2 x_4, c_1 x_2^2 - x_1^2, c_3 x_3 x_5^2 - x_1^3, c_1 x_1^3 - c_3 x_4 x_5^2 \rangle. \qquad (4.4) $$
<figure><img src="image_5.png" /><figcaption>Fig. 1. The Whitney umbrella $W \subset \mathbb{R}^3$ obtained from (4.6); it contains the real toric variety $Y = W \cap \mathbb{R}_{\ge 0}^3$.</figcaption></figure>

$$ \pi : \mathbb{V}(I_5) \to \mathbb{C}_c^7, \quad \mathbb{V}(I_5) \subset \mathbb{C}^{16}, $$ 
$$ \begin{aligned} I_5 = & (c_3x_5 - c_2x_7, c_5x_4 - c_4x_6, c_3x_3x_9 - c_6x_7, c_2x_3x_9 - c_6x_5, \\ & c_5x_2x_9 - c_7x_6, c_4x_2x_9 - c_7x_4, c_6x_2x_8 - c_6x_5, \\ & c_3x_2x_8 - c_2x_7, c_2x_2x_8 - c_2x_5, c_7x_1x_8 - c_1x_2x_9, \\ & c_5x_1x_8 - c_1x_6, c_4x_1x_8 - c_1x_4, c_6c_7x_6x_8 - c_5c_6x_5x_9, \\ & c_3c_7x_6x_8 - c_2c_5x_7x_9, c_2c_7x_6x_8 - c_2c_5x_5x_9, \\ & c_6c_7x_4x_8 - c_4c_6x_5x_9, c_3c_7x_4x_8 - c_2c_4x_7x_9, \\ & c_2c_7x_4x_8 - c_2c_4x_5x_9, c_3c_7x_3x_6 - c_5c_6x_2x_7, \\ & c_1c_3x_2x_6 - c_2c_5x_1x_7, c_5c_6x_2x_5 - c_2c_7x_3x_6, \\ & c_5c_6x_1x_5 - c_1c_6x_2x_6, c_2c_5x_1x_5 - c_1c_2x_2x_6, \\ & c_3c_7x_3x_4 - c_4c_6x_2x_7, c_2c_7x_3x_4 - c_4c_6x_2x_5, \\ & c_1c_6x_2x_4 - c_4c_6x_1x_5, c_1c_3x_2x_4 - c_2c_4x_1x_7, \\ & c_1c_2x_2x_4 - c_2c_4x_1x_5, c_1c_6x_2^2 - c_2c_7x_1x_3). \end{aligned} \quad (4.5) $$

## 4.2. Concluding example

We conclude with a more detailed example to illustrate the various steps of the algorithm. Let $k \in \{\mathbb{R}_{\ge 0}, \mathbb{C}\}$. Consider a family $\mathcal{C} := \{C_c\}_{c \in k} \subset k^2$ of plane curves given by the vanishing locus of ideals of the form

$$ c y^2 - x^2. \quad (4.6) $$

Then, the curve $C_c$ is diffeomorphic to $\{x = \pm y\} \cap k^2$, for any $c \neq 0$ in $k$, while $C_0$ has no other representative in $\mathcal{C}$.

This observation can be confirmed from a different point of view: The family $\mathcal{C}$ consists of fibers $\pi^{-1}|_Y(c)$ under the map $\pi : Y \to k$, $\pi : (x, y, c) \mapsto c$, where $Y \subset k^3$ is the vanishing locus of the ideal (4.6) in $\mathbb{Z}[x, y, c]$ (see Fig. 1). The latter is a toric ideal, whose $A$-matrix is

$$ A := \begin{bmatrix} 1 & 0 & 2 \\ 1 & 1 & 0 \end{bmatrix} = [a_1 \ a_2 \ a_3]. $$

We can decompose $Y$ into seven strata, each of which corresponds to a face of the polytope $P \subset \mathbb{R}^2$ corresponding to $A$, (see Fig. 2). This polytope has 3 vertices, which are the columns $a_1, a_2, a_3$ of $A$, and three edges $[a_3, a_1] = \text{conv}(a_3, a_1)$, $[a_2, a_1] = \text{conv}(a_2, a_1)$, and $[a_2, a_3] = \text{conv}(a_2, a_3)$. The
<figure><img src="image_6.png" /><figcaption>Fig. 2. The polytope P corresponding to A and the variety Y.</figcaption></figure>

collection of edges $[a_3, a_1]$, $[a_2, a_1]$, and $[a_2, a_3]$, gives rise to the following affine toric varieties (which are all copies of $k$):

$$X_{[a_3, a_1]} := Y \cap \{y = 0\} = \mathbb{V}(x, y),$$

$$X_{[a_2, a_1]} := Y \cap \{c = 0\} = \mathbb{V}(x, c),$$

$$X_{[a_2, a_3]} := Y \cap \{x = 0\} = \mathbb{V}(x, y) \cup \mathbb{V}(x, c).$$

On the codomain side, the edge $[a_3, a_1]$ gives a copy of $Z = k$ (as does the whole polytope P), the edge $[a_2, a_1]$ gives a copy of $\mathbb{V}(c) \subset Z$, and the edge $[a_2, a_3]$ gives another copy of $Z = k$. The collection of vertices $a_1, a_2$, and $a_3$, give rise to

$$X_{a_1} := Y \cap \{y = c = 0\} = \mathbb{V}(x, y, c) = \{(0, 0, 0)\},$$

$$X_{a_2} := Y \cap \{x = c = 0\} = \mathbb{V}(x, c),$$

$$X_{a_3} := Y \cap \{x = y = 0\} = \mathbb{V}(x, y).$$

On the codomain side, $a_1$ and $a_2$ give copies of $\mathbb{V}(c) \subset Z$, while $a_3$ gives a copy of $Z$.

Hence, the stratification $(Y_\bullet, Z_\bullet)$ of $\pi$ is given by $Y_\bullet = (Y_2, Y_1, Y_0)$, $Z_\bullet = (Z_1, Z_0)$, where $Y_2 = X_A = Y_1$,

$$Y_1 = X_{[a_3, a_1]} \cup X_{[a_2, a_1]} \cup X_{[a_2, a_3]} \cup X_{a_2} \cup X_{a_3} = \mathbb{V}(x, y) \cup \mathbb{V}(x, c),$$

$X_0 = X_{a_1}$, $Z_1 = Z = k$, and $Z_0 = \mathbb{V}(c) = 0 \subset k$. Therefore, in particular, we obtain the stratification $Z_\bullet = \{\emptyset \subset \{0\} \subset k\}$ of $k$. Our main result states that over each connected component of $k^* = k - 0$ the map $\pi$ is a fibrated stratification; this means that all fibers over non-zero points are diffeomorphic. Hence, the curves $C_c$, as in (4.6), are diffeomorphic to $\{x = \pm y\} \cap k^2$ for all $c \neq 0$ in $k$.

CRediT authorship contribution statement

**Boulos El Hilany:** Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing.
**Martin Helmer:** Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing.

**Elias Tsigaridas:** Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing.
Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relation-
ships that could have appeared to influence the work reported in this paper.

The author is an Editorial Board Member/Editor-in-Chief/Associate Editor/Guest Editor for this jour-
nal and was not involved in the editorial review or the decision to publish this article.

Data availability

No data was used for the research described in the article.

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