arXiv subjects

Askold Khovanskii

Publications and source records attributed to Askold Khovanskii.

At least 19 recordsLinked to original sources

arXiv2026

Polynomial valuations on plane polygons

Scissors congruence problems involving translations have prompted the study of translation invariant simple valuations. We review this classical theory from a naive and consistent viewpoint: starting from a description of all simple valuations on polygons, we characterize the effect of translation invariance. A description of all polynomial simple valuations is obtained as a bi-product of the adopted approach and as a direct generalization of the translation invariant theory; it appears to be new.

math.MG↗

arXiv2026

Galois theory for finite fields

This note presents Galois theory for finite fields. It was written as a handout for the MAT401 course ``Polynomial equations and fields'' taught at the University of Toronto in Spring 2026. We use without proofs some basic properties of finite fields and of finite field extensions which we already covered in class. Firstly, we describe an extension $K\subset F$ of a finite field $K$ of a given degree $n$. We show that the set of all intermediate fields for this extension is in one-to-one correspondence with the set of all divisors $k$ of the degree $n$. Then we describe the Galois group of this extension which is the cyclic group of order $n$. The set of subgroups of this group also is in one-to-one correspondence with the set of all divisors $k$ of the degree $n$. It allows us to prove the Galois correspondence for that extension. In the last section, we state basic theorems of Galois theory for arbitrary fields which will be proven later in the course.

math.NT↗

arXiv2026

A necessary condition for solvability by radicals

This note was prepared as a handout for the MAT401 course ``Polynomial equations and fields", taught at the University of Toronto in Spring 2026. It presents a proof of a necessary condition for the solvability of algebraic equations by radicals, based on Galois theory. We begin with a brief overview of the relevant basic results from Galois theory, as covered in MAT401, and use -- without proof -- several standard (and relatively simple) results from the course textbook [1]. The sufficient condition for solvability by radicals, which is based on linear algebra, we will present in the next handout.

math.RA↗

arXiv2026

Valuations on polyhedra and topological arrangements

We revisit a classical theme of (general or translation invariant) valuations on convex polyhedra. Our setting generalizes the classical one, in a ``dual'' direction to previously considered generalizations: while previous research was mostly concerned with variations of ground fields/rings, over which the vertices of polytopes are defined, we consider more general collections of defining hyperplanes. No algebraic structures are imposed on these collections. This setting allows us to uncover a close relationship between scissors congruence problems on the one hand and finite hyperplane arrangements on the other hand: there are many parallel results in these fields, for which the parallelism seems to have gone unnoticed. In particular, certain properties of the Varchenko--Gelfand algebras for arrangements translate to properties of polytope rings or valuations. Studying these properties is possible in a general topological setting, that is, in the context of the so-called topological arrangements, where hyperplanes do not have to be straight and may even have nontrivial topology.

math.CO↗

arXiv2025

Semigroups, Cartier divisors and convex bodies

The theory of Newton--Okounkov bodies provides direct relations and points out analogies between the theory of mixed volumes of convex bodies, on the one hand, and the intersection theories of Cartier divisors and of Shokurov $b$-divisors, on the other hand. The classical inequalities between mixed volumes of convex bodies correspond to inequalities between intersection indices of nef Cartier divisors on an irreducible projective variety and between the birationally invariant intersection indices of nef type Shokurov $b$-divisors on an irreducible algebraic variety. Such algebraic inequalities are known as Khovanskii--Teissier inequalities. Our proof of these inequalities is based on simple geometric inequalities on two dimensional convex bodies and on pure algebraic arguments. The classical geometric inequalities follow from the algebraic inequalities. We collected results from a few papers which were published during the last forty five years. Some theorems of the present paper were never stated, but all ideas needed for their proofs are contained in the published papers. So we avoid all heavy proofs. Our goal is to present an overview of the area. Notions of homogeneous polynomials on commutative semigroups and of their polarizations provide an adequate language for discussing the subject. We use this language through the paper.

math.AG↗

arXiv2025

Vector-valued Laurent polynomial equations, toric vector bundles and matroids

Let $L \subset \mathbb{C}^r \otimes \mathbb{C}[x_1^\pm, \ldots, x_n^\pm]$ be a finite dimensional subspace of vector-valued Laurent polynomials invariant under the action of torus $(\mathbb{C}^*)^n$. We study subvarieties in the torus, defined by equations $f = 0$ for generic $f \in L$. We generalize the BKK theorem, that counts the number of solutions of a system of Laurent polynomial equations generic for their Newton polytopes, to this setting. The answer is in terms of mixed volume of certain virtual polytopes encoding discrete invariants of $L$ which involves matroid data. Moreover, we prove an Alexandrov-Fenchel type inequality for these virtual polytopes. Finally, we extend this inequality to non-representable polymatroids. This extends the usual Alexandrov-Fenchel inequality for polytopes as well as log-concavity results related to matroids.

math.AG↗

arXiv2025

On solvability of linear differential equations in finite terms

We consider the problem of solvability of linear differential equations over a differential field~$K$. We introduce a class of special differential field extensions, which widely generalizes the classical class of extensions of differential fields by integrals and by exponentials of integrals and which has similar properties. We announce the following result: if a linear differential equation over $K$ can not be solved by generalized quadratures, then no special extension can help solve it. In the paper we prove a weaker version of this result in which we consider only pure transcendental extensions of $K$. Our paper is self-contained and understandable for beginners. It demonstrates the power of Liouville's original approach to problems of solvability of equations in finite terms.

math.AG↗

arXiv2023

Center of Mass Technique and Affine Geometry

The notion of center of mass, which is very useful in kinematics, proves to be very handy in geometry (see [1]-[2]). Countless applications of center of mass to geometry go back to Archimedes. Unfortunately, the center of mass cannot be defined for sets whose total mass equals zero. In the paper we improve this disadvantage and assign to an n-dimensional affine space L over any field k the (n+1)-dimensional vector space over the field k of weighty points and mass dipoles in L. In this space, the sum of weighted points with nonzero total mass is equal to the center of mass of these points equipped with their total mass. We present several interpretations of the space of weighty points and mass dipoles in L, and a couple of its applications to geometry. The paper is self-contained and is accessible for undergraduate students.

math.MG↗

arXiv2023

Fibered toric varieties

A toric variety is called fibered if it can be represented as a total space of fibre bundle over toric base and with toric fiber. Fibered toric varieties form a special case of toric variety bundles. In this note we first give an introduction to the class of fibered toric varieties. Then we use them to illustrate some known and conjectural results on topology and intersection theory of general toric variety bundles. Finally, using the language of fibered toric varieties, we compute the equivariant cohomology rings of smooth complete toric varieties.

math.AG↗

arXiv2022

Total order compatible with addition on commutative semigroups

In the paper, we describe all total orders $\succ$ compatible with addition on additive subsemigroup $S$ of finite dimensional spaces over rational numbers. We provide a necessary and sufficient condition under which a finitely generated semigroups $S$ equipped with an order $\succ$ is a well-ordered set. We also present some auxiliary results on orders compatible with addition on additive subsemigroups of finite dimensional spaces over real numbers. All arguments in this paper are based on two simple theorems in the geometry of convex (not necessarily closed) sets. Proofs of these theorems are presented for readers' convenience. A first version of this paper was written as a handout for my graduate course on the theory of Newton--Okounkov bodies.

math.AG↗

arXiv2022

Generalized virtual polytopes and quasitoric manifolds

In this paper we develop a theory of volume polynomials of generalized virtual polytopes based on the study of topology of affine subspace arrangements in a real Euclidean space. We apply this theory to obtain a topological version of the BKK Theorem, the Stanley-Reisner and Pukhlikov-Khovanskii type descriptions for cohomology rings of generalized quasitoric manifolds.

math.AT↗

arXiv2022

Interpolation Polynomials and Linear Algebra

We reconsider the theory of Lagrange interpolation polynomials with multiple interpolation points and apply it to linear algebra. For instance, $A$ be a linear operator satisfying a degree $n$ polynomial equation $P(A)=0$. One can see that the evaluation of a meromorphic function $F$ at $A$ is equal to $Q(A)$, where $Q$ is the degree $<n$ interpolation polynomial of $F$ with the the set of interpolation points equal to the set of roots of the polynomial $P$. In particular, for $A$ an $n \times n$ matrix, there is a common belief that for computing $F(A)$ one has to reduce $A$ to its Jordan form. Let $P$ be the characteristic polynomial of $A$. Then by the Cayley-Hamilton theorem, $P(A)=0$. And thus the matrix $F(A)$ can be found without reducing $A$ to its Jordan form. Computation of the Jordan form for $A$ involves many extra computations. In the paper we show that it is not needed. One application is to compute the matrix exponential for a matrix with repeated eigenvalues, thereby solving arbitrary order linear differential equations with constant coefficients.

math.CA↗

arXiv2021

Cohomology rings of quasitoric bundles

The classical BKK theorem computes the intersection number of divisors on toric variety in terms of volumes of corresponding polytopes. It was observed by Pukhlikov and the first author that the BKK theorem leads to a presentation of the cohomology ring of toric variety as a quotient of the ring of differential operators with constant coefficients by the annihilator of an explicit polynomial. In this paper we generalize this construction to the case of quasitoric bundles. These are fiber bundles with generalized quasitoric manifolds as fibers. First we obtain a generalization of the BKK theorem to this case. Then we use recently obtained descriptions of the graded-commutative algebras which satisfy Poincaré duality to give a description of cohomology rings of quasitoric bundles.

math.AT↗

arXiv2021

Gorenstein algebras and toric bundles

We study commutative algebras with Gorenstein duality, i.e. algebras $A$ equipped with a non-degenerate bilinear pairing such that $\langle ac,b\rangle=\langle a,bc\rangle$ for any $a,b,c\in A$. If an algebra $A$ is Artinian, such pairing exists if and only if $A$ is Gorenstein. We give a description of algebras with Gorenstein duality as a quotients of the ring of differential operators by the annihilator of an explicit polynomial (or more generally formal polynomial series). This provides a calculation of Macaulay's inverse systems for (not necessarily Artinian) algebras with Gorenstein duality. Our description generalizes previously know description of graded algebras with Gorenstein duality generated in degree 1. Our main motivation comes from the study of even degree cohomology rings. In particular, we apply our main result to compute the ring of cohomology classes of even degree of toric bundles and the ring of conditions of horospherical homogeneous spaces.

math.AC↗

arXiv2021

Cohomology rings of toric bundles and the ring of conditions

The celebrated BKK Theorem expresses the number of roots of a system of generic Laurent polynomials in terms of the mixed volume of the corresponding system of Newton polytopes.Pukhlikov and the second author noticed that the cohomology ring of smooth projective toric varieties over $\mathbb{C}$ can be computed via the BKK Theorem. This complemented the known descriptions of the cohomology ring of toric varieties, like the one in terms of Stanley-Reisner algebras. Sankaran and Uma generalized the "Stanley-Reisner description" to the case of toric bundles, i.e. equivariant compactifications of (not necessarily algebraic) torus principal bundles. We provide a description of the cohomology ring of toric bundles which is based on a generalization of the \BKK Theorem, and thus extends the approach by Pukhlikov and the second author. Indeed, for every cohomology class of the base of the toric bundle, we obtain a BKK-type theorem. Furthermore, our proof relies on a description of graded-commutative algebras which satisfy Poincaré duality. From this computation of the cohomology ring of toric bundles, we obtain a description of the ring of conditions of horospherical homogeneous spaces as well as a version of Brion-Kazarnovskii theorem for them. We conclude the manuscript with a number of examples. In particular, we apply our results to toric bundles over a full flag variety $G/B$. The description that we get generalizes the corresponding description of the cohomology ring of toric varieties as well as the one of full flag varieties $G/B$ previously obtained by Kaveh.

math.AG↗

arXiv2020

Newton polyhedra and good compactification theorem

A new transparent proof of the well known good compactification theorem for the complex torus $(\Bbb C^*)^n$ is presented. This theorem provides a powerful tool in enumerative geometry for subvarieties in the complex torus. The paper also contains an algorithm constructing a good compactification for a subvariety in $(\Bbb C^*)^n$ explicitly defined by a system of equations. A new theorem on a torodoidal like compactification is stated. A transparent proof of this generalization of the good compactification theorem which is similar to proofs and constructions from this paper will be presented in a forthcoming publication.

math.AG↗

arXiv2019

Comments on J. F. Ritt's book "Integration in Finite Terms"

The First and Second Liouville's Theorems provide correspondingly criterium for integrability of elementary functions "in finite terms" and criterium for solvability of second order linear differential equations by quadratures. The brilliant book of J.F.~Ritt contains proofs of these theorems and many other interesting results. This paper was written as comments on the book but one can read it independently. The first part of the paper contains modern proofs of The First Theorem and of a generalization of the Second Theorem for linear differential equations of any order. In the second part of the paper we present an outline of topological Galois theory which provides an alternative approach to the problem of solvability of equations in finite terms. The first section of this part deals with a topological approach to representability of algebraic functions by radicals and to the 13-th Hilbert problem. This section is written with all proofs. Next sections contain only statements of results and comments on them (basically no proofs are presented there).

math.AG↗

arXiv2019

On Representability of Algebraic Functions by Radicals

This preprint is dedicated to a self contained simple proof of the classical criteria for representability of algebraic functions of several complex variables by radicals. It also contains a criteria for representability of algebroidal functions by composition of single-valued analytic functions and radicals, and a result related to the 13-th Hilbert problem. This preprint is an extended version of the author's 1971 paper. It is written as a part of the comments to a new edition (in preparation) of the classical book `Integration in finite terms' by J.F.~Ritt.

math.AG↗