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Alexander Esterov

Publications and source records attributed to Alexander Esterov.

At least 19 recordsLinked to original sources
arXiv2026

Engineered Complete Intersections: Algorithmic Aspects

Engineered Complete Intersections (ECI's) are a class of sparse polynomial systems frequently arising in a number of contexts, both in pure mathematics (e.g. enumerative geometry) and applications (e.g. chemical reaction networks). Based on theoretical results given by the first author, we give several contributions. First we give a new effective technique to tropicalize such systems by generalizing the classical notion of mixed subdivisions introduced by Huber and Sturmfels (1995) to ECI's with the particular goal to efficiently count solutions of square systems of equations in ECI form. We further design a tropical homotopy continuation algorithm for computing such mixed subdivisions, inspired by Jensen (2016), Malajovich (2017) and Daisey and Ren (2024). Our techniques can be used to numerically solve such systems by coupling them with the algorithms introduced by Helminck, Henriksson and Ren (2024). Finally, we give an algorithm to compute Newton polytopes of eliminants of ECI's. This gives a new way to compute, for example, Newton polytopes of so-called $A$-discriminants. Coupled with evaluation-interpolation paradigms our algorithm gives an efficient approach to compute such eliminants. We implemented our algorithms in the form of a software package which we use to demonstrate their practical feasibility on a range of examples.

cs.SC↗
arXiv2025

Bernstein-Kouchnirenko-Khovanskii with a symmetry

A generic polynomial f(x,y,z) with a prescribed Newton polytope defines a symmetric spatial curve f(x,y,z)=f(y,x,z)=0. We study its geometry: the number, degree and genus of its irreducible components, the number and type of singularities, etc. and discuss to what extent these results generalize to higher dimension and more complicated symmetries. As an application, we characterize generic one-parameter families of complex univariate polynomials, whose Galois group is a complete symmetric group.

math.AG↗
arXiv2025

Engineered complete intersections: eliminating variables and understanding topology

We continue the study of engineered complete intersections (ECI) -- an umbrella generality for a number of important objects in combinatoiral and applied algebraic geometry (such as nondegenerate toric complete intersections, critical loci of their projections, hyperplane arrangements, generalized Calabi--Yau complete intersections, incidence varieties in algebraic optimization, reaction networks). In this paper, we work on extending to ECIs several classical results about toric complete intersections. This includes elimination theory, patchworking over ${\mathbb R}$, and computing basic geometric invariants over ${\mathbb C}$. Our results apply e.g. to eliminating variables in systems of ODEs, such as reaction networks, computing Newton polytopes of discriminants, constructing real polynomial maps and reaction networks with prescribed topology. Along the way, we assign a cohomology ring to an arbitrary tropical fan, and relate reducible ECIs to arrangements of pairwise intersecting planes.

math.AG↗
arXiv2024

Basecondary polytopes

Many (if not most) of convex polytopes, important for combinatorial and algebraic geometry, are closely related to secondary polytopes of point configurations, or base polytopes of submodular functions, or their numerous variations and generalizations. The aim of this text is to introduce the class of basecondary polytopes. This class includes (and allows to study uniformly) the aforementioned ones, as well as some others, e.g. appearing as Newton polytopes of important discriminant hypersurfaces. Most notably, this includes the discriminant of the Lyashko--Looijenga map, which is important for enumerative geometry of ramified coverings and cannot be reduced (by far) to Gelfand--Kapranov--Zelevinsky's A-discriminants and secondary polytopes.

math.CO↗
arXiv2024

Gauss--Bonnet for CSM classes of very affine manifolds, and tropical isotopies

The CSM class of a very affine manifold $U$ is represented by the rank drop locus of a general tuple of torus invariant 1-forms on it. This equality holds in the homology of any toric compactification $X\supset U$. It was proved for schön $U$ by Huh, and later for all $U$ in the homology of $X=CP^n$ by Maxim--Rodriguez--Wang--Wu, using Ginsburg's interpretation of CSM classes as Lagrangian cycles. We deduce this identity in full generality from properties of affine characteristic classes, give an explicit sufficient condition of general position for the 1-forms, and use it to extend the identity to non-torus invariant 1-forms. Along the way, we characterize very affine varieties of ML degree 0, and give a useful criterion for a family of varieties to have a constant tropical fan (``tropical isotopy theorem''). Our main idea is a technique to study very affine varieties without compactifying them. As a result we do not have to deal with singularities occuring at the boundary of the compactification, which would require less constructive methods (such as resolution of singularities). This is what enables us to give constructive genericity conditions in this Gauss--Bonnet theorem.

math.AG↗
arXiv2024

Schön complete intersections

A complete intersection $f_1=\cdots=f_k=0$ is schön, if $f_1=\cdots=f_j=0$ defines a schön subvariety of an algebraic torus for every $j\leqslant k$. This class includes nondegenerate complete intersections, critical loci of their coordinate projections, other simplest Thom--Boardman and multiple point strata of such projections, generalized Calabi--Yau complete intersections, equaltions of polynomial optimization, hyperplane arrangement complements, and many other interesting special varieties. We study their Euler characteristics, connectednes, Calabi--Yau-ness, tropicalizations, etc., extending (in part conjecturally) the respective classical results about nondegenerate complete intersections.

math.AG↗
arXiv2024

Engineered complete intersections: slightly degenerate Bernstein--Kouchnirenko--Khovanskii

Geometry of sparse systems of polynomial equations (i.e. the ones with prescribed monomials and generic coefficients) is well studied in terms of their Newton polytopes. The results of this study are colloquially known as the Bernstein--Kouchnirenko--Khovanskii toolkit, and unfortunately are not applicable to many important systems, whose coefficients slightly fail to be generic. This for instance happens if some of the equations are obtained from another one by taking partial derivatives or permuting the variables, or the equations are linear, realizing a non-trivial matroid, or in more advanced settings such as generalized Calabi--Yau complete intersections. Such interesting examples (as well as many others) turn out to belong to a natural class of ``systems of equations that are nondegenerate upon cancellations''. We extend to this class several classical and folklore results of the Bernstein--Kouchnirenko--Khovanskii toolkit, such as the ones regarding the number and regularity of solutions, their irreducibility, tropicalization and Calabi--Yau-ness.

math.AG↗
arXiv2023

Sparse curve singularities, singular loci of resultants, and Vandermonde matrices

We compute the $δ$-invariant of a curve singularity parameterized by generic sparse polynomials. We apply this to describe topological types of generic singularities of sparse resultants and ``algebraic knot diagrams'' (i.e. generic algebraic spatial curve projections). Our approach is based on some new results on zero loci of Schur polynomials, on transversality properties of maps defined by sparse polynomials, and on a new refinement of the notion of tropicalization of a curve (ultratropicalization), which may be of independent interest.

math.AG↗
arXiv2022

Permuting the roots of univariate polynomials whose coefficients depend on parameters

We address two interrelated problems concerning the permutation of roots of univariate polynomials whose coefficients depend on parameters. First, we compute the Galois group of polynomials $\varphi(x)\in\mathbb{C}[y_1,\cdots,y_k][x]$ over $\mathbb{C}(y_1,\cdots,y_k)$. Provided that the corresponding multivariate polynomial $\varphi(x,y_1,\ldots,y_k)$ is generic with respect to its support $A\subset \mathbb{Z}^{k+1}$, we determine the associated Galois group for any such $A$. Second, we determine the Galois group of systems of polynomial equations of the form $p(x,y)=q(y)=0$ where $p$ and $q$ have fixed supports $A_1\subset \mathbb{Z}^2$ and $A_2\subset \{0\}\times \mathbb{Z}$, respectively. For each problem, we determine the image of an appropriate braid monodromy map in order to compute the sought Galois group. Among the applications, we determine the Galois group of any rational function generic with respect to its support. We also provide general obstructions to the Galois group of enumerative problems over algebraic groups.

math.AG↗
arXiv2022

On the monodromy conjecture for non-degenerate hypersurfaces

The monodromy conjecture is an umbrella term for several conjectured relationships between poles of zeta functions, monodromy eigenvalues and roots of Bernstein-Sato polynomials in arithmetic geometry and singularity theory. Even the weakest of these relations -- the Denef--Loeser conjecture on topological zeta functions -- is open for surface singularities. We prove it for a wide class of multidimensional singularities that are non-degenerate with respect to their Newton polyhedra, including all such singularities of functions of four variables. A crucial difference from the known case of three variables is the existence of degenerate singularities arbitrarily close to a non-degenerate one. Thus, even aiming at the study of non-degenerate singularities, we have to go beyond this setting. We develop new tools to deal with such multidimensional phenomena, and conjecture how the proof for non-degenerate singularities of arbitrarily many variables might look like.

math.AG↗
arXiv2021

The ring of local tropical fans and tropical nearby monodromy eigenvalues

We extend the tropical intersection theory to tropicalizations of germs of analytic sets. In particular, we construct a (not entirely obvious) local version of the ring of tropical fans with a nondegenerate intersection pairing. As an application, we study nearby monodromy eigenvalues -- the eigenvalues of the monodromy operators of singularities, adjacent to a given singularity of a holomorphic function $f$. More precisely, we express some of such values in terms of certain resolutions of $f$. The expression is given it terms of the exceptional divisor strata of arbitrary codimension, generalizing the classical A'Campo formula that consumes only codimension 1 strata and produces only monodromy eigenvalues at the origin. For this purpose, we introduce tropical characteristic classes of germs of analytic sets, and use this calculus to detect some of the nearby monodromy eigenvalues, which we call tropical. The study is motivated by the monodromy conjectures by Igusa, Denef and Loeser: every pole of an appropriate local zeta function of $f$ induces a nearby monodromy eigenvalue. We propose a presumably stronger version of this conjecture: all poles of the local zeta function induce tropical nearby monodromy eigenvalues. In particular, if the singularity is non-degenerate with respect to its Newton polyhedron $N$, then the tropical monodromy eigenvalues can be expressed in terms of fiber polytopes of certain faces of $N$, so our conjecture (unlike the original ones) becomes a purely combinatorial statement about a polyhedron. This statement is confirmed for the topological zeta function in dimension up to 4 in a joint work with A. Lemahieu and K. Takeuchi, which, in particular, supports our conjecture and proves the original one for non-degenerate singularities in 4 variables.

math.AG↗
arXiv2020

Galois theory for general systems of polynomial equations

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables. In particular, our result proves the multivariate version of the Abel--Ruffini theorem: the classification of general systems of equations solvable by radicals reduces to the classification of lattice polytopes of mixed volume 4 (which we prove to be finite in every dimension). We also notice that the monodromy of every general system of equations is either symmetric or imprimitive, similarly to what Sottile and White conjectured in Schubert calculus. The proof is based on a new result of independent importance regarding dual defectiveness of systems of equations: the discriminant of a reduced irreducible square system of general polynomial equations is a hypersurface unless the system is linear up to a monomial change of variables.

math.AG↗
arXiv2020

Braid monodromy of univariate fewnomials

Let $\mathcal{C}_d\subset \mathbb{C}^{d+1}$ be the space of non-singular, univariate polynomials of degree $d$. The Viète map $\mathscr{V} : \mathcal{C}_d \rightarrow Sym_d(\mathbb{C})$ sends a polynomial to its unordered set of roots. It is a classical fact that the induced map $\mathscr{V}_*$ at the level of fundamental groups realises an isomorphism between $π_1(\mathcal{C}_d)$ and the Artin braid group $B_d$. For fewnomials, or equivalently for the intersection $\mathcal{C}$ of $\mathcal{C}_d$ with a collection of coordinate hyperplanes in $\mathbb{C}^{d+1}$, the image of the map $\mathscr{V} _* : π_1(\mathcal{C}) \rightarrow B_d$ is not known in general. In the present paper, we show that the map $\mathscr{V} _*$ is surjective provided that the support of the corresponding polynomials spans $\mathbb{Z}$ as an affine lattice. If the support spans a strict sublattice of index $b$, we show that the image of $\mathscr{V} _*$ is the expected wreath product of $\mathbb{Z}/b\mathbb{Z}$ with $B_{d/b}$. From these results, we derive an application to the computation of the braid monodromy for collections of univariate polynomials depending on a common set of parameters.

math.AG↗
arXiv2020

Sparse polynomial equations and other enumerative problems whose Galois groups are wreath products

We introduce a new technique to prove connectivity of subsets of covering spaces (so called inductive connectivity), and apply it to Galois theory of problems of enumerative geometry. As a model example, consider the problem of permuting the roots of a complex polynomial $f(x) = c_0 + c_1 x^{d_1} + \ldots + c_k x^{d_k}$ by varying its coefficients. If the GCD of the exponents is $d$, then the polynomial admits the change of variable $y=x^d$, and its roots split into necklaces of length $d$. At best we can expect to permute these necklaces, i.e. the Galois group of $f$ equals the wreath product of the symmetric group over $d_k/d$ elements and $\mathbb{Z}/d\mathbb{Z}$. The aim of this paper is to prove this equality and study its multidimensional generalization: we show that the Galois group of a general system of polynomial equations equals the expected wreath product for a large class of systems, but in general this expected equality fails, making the problem of describing such Galois groups unexpectedly rich.

math.AG↗
arXiv2019

Confluent A-hypergeometric functions and rapid decay homology cycles

We study confluent A-hypergeometric functions introduced by Adolphson. In particular, we give their integral representations by using rapid decay homology cycles of Hien and obtain a formula for the asymptotic expansions at infinity of confluent A-hypergeometric functions.

math.AG↗
arXiv2016

Signs of the Leading Coefficients of the Resultant

We construct a certain $\F_2$-valued analogue of the mixed volume of lattice polytopes. This 2-mixed volume cannot be defined as a polarization of any kind of an additive measure, or characterized by any kind of its monotonicity properties, because neither of the two makes sense over $\F_2$. In this sense, the convex-geometric nature of the 2-mixed volume remains unclear. On the other hand, the 2-mixed volume seems to be no less natural and useful than the classical mixed volume -- in particular, it also plays an important role in algebraic geometry. As an illustration of this role, we obtain a closed-form expression in terms of the 2-mixed volume to compute the signs of the leading coefficients of the resultant, which were by now explicitly computed only for some special cases.

math.AG↗
arXiv2016

Characteristic classes of affine varieties and Plucker formulas for affine morphisms

An enumerative problem on a variety $V$ is usually solved by reduction to intersection theory in the cohomology of a compactification of $V$. However, if the problem is invariant under a "nice" group action on $V$ (so that $V$ is spherical), then many authors suggested a better home for intersection theory: the direct limit of the cohomology rings of all equivariant compactifications of $V$. We call this limit the affine cohomology of $V$ and construct affine characteristic classes of subvarieties of a complex torus, taking values in the affine cohomology of the torus. This allows us to make the first steps in computing affine Thom polynomials. Classical Thom polynomials count how many fibers of a generic proper map of a smooth variety have a prescribed collection of singularities, and our affine version addresses the same question for generic polynomial maps of affine algebraic varieites. This notion is also motivated by developing an intersection-theoretic approach to tropical correspondence theorems: they can be reduced to the computation of affine Thom polynomials, because the fundamental class of a variety in the affine cohomology is encoded by the tropical fan of this variety. The first concrete answer that we obtain is the affine version of what were, historically speaking, the first three Thom poylnomials -- the Plucker formulas for the degree and the number of cusps and nodes of a projectively dual curve. This, in particular, classifies toric varieties, whose projective dual is a hypersurface, computes the tropical fan of the variety of double tangent hyperplanes to a toric variety, and describes the Newton polytope of the hypersurface of non-Morse polynomials of a given degree. We also make a conjecture on the general form of affine Thom polynomials -- a key ingredient is the $n$-ary fan, generalizing the secondary polytope.

math.AG↗