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Askold Khovanskii

Publications and source records attributed to Askold Khovanskii.

30 records · Page 2Linked to original sources
arXiv2019

Multidimensional topological Galois theory

In this preprint we present an outline of the multidimensional version of topological Galois theory. The theory studies topological obstruction to solvability of equations "in finite terms" (i.e. to their solvability by radicals, by elementary functions, by quadratures and so on). This preprint is based on the author's book on topological Galois theory. It contains definitions, statements of results and comments to them. Basically no proofs are presented. This preprint was 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.

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arXiv2019

One dimensional topological Galois theory

In the preprint we present an outline of the one dimensional version of topological Galois theory. The theory studies topological obstruction to solvability of equations "in finite terms" (i.e. to their solvabilty by radicals, by elementary functions, by quadratures and so on). The preprint is based on the author's book on topological Galois theory. It contains definitions, statements of results and comments to them. Basically no proofs are presented. The preprint was 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. }

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arXiv2018

Solvability of Equations by Quadratures and Newton's Theorem

Picard--Vessiot theorem (1910) provides a necessary and sufficient condition for solvability of linear differential equations of order $n$ by quadratures in terms of its Galois group. It is based on the differential Galois theory and is rather involved. J.Liouville in 1839 found an elementary criterium for such solvability for $n=2$. J.F.Ritt simplified Liouville's theorem (1948). In 1973 M. Rosenlicht proved a similar criterium for arbitrary $n$. Rosenlicht work relies on the valuation theory and is not elementary. In these notes we show that the elementary Liouville--Ritt method based on developing solutions in Puiseux series as functions of a parameter works smoothly for arbitrary $n$ and proves the same criterium.

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arXiv2018

Integrability in Finite Terms And Actions of Lie Groups

According to Liouville's Theorem, an indefinite integral of an elementary function is usually not an elementary function. In this notes, we discuss that statement and a proof of this result. The differential Galois group of the extension obtained by adjoining an integral does not determine whether the integral is an elementary function or not. Nevertheless, Liouville's Theorem can be proved using differential Galois groups. The first step towards such a proof was suggested by Abel. This step is related to algebraic extensions and their finite Galois groups. A significant part of this notes is dedicated to a second step, which deals with pure transcendent extensions and their Galois groups which are connected Lie groups. The idea of the proof goes back to J.Liouville and J.Ritt.

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arXiv2017

Newton polyhedra, tropical geometry and the ring of condition for $(C^*)^n$

The ring of conditions defined by C. De Concini and C. Procesi is an intersection theory for algebraic cycles in a spherical homogeneous space. In the paper we consider the ring of conditions for the group $(C^*)^n$. Up to a big extend this ring can be reduced to the cohomology rings of smooth projective toric varieties. This ring also can be described using tropical geometry. We recall these known results and provide a new description of this ring in terms of convex integral polyhedra.

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arXiv2017

The Resultant of Developed Systems of Laurent Polynomials

Let $R_Δ(f_1,\ldots,f_{n+1})$ be the {\it $Δ$-resultant} (see below) of $(n+1)$-tuple of Laurent polynomials. We provide an algorithm for computing $R_Δ$ assuming that an $n$-tuple $(f_2,\dots,f_{n+1})$ is {\it developed} (see sec.6). We provide a relation between the product of $f_1$ over roots of $f_2=\dots=f_{n+1}=0$ in $(\mathbb C^*)^n$ and the product of $f_2$ over roots of $f_1=f_3=\dots=f_{n+1}=0$ in $(\mathbb C^*)^n$ assuming that the $n$-tuple $(f_1f_2,f_3,\ldots,f_{n+1})$ is developed. If all $n$-tuples contained in $(f_1,\dots,f_{n+1})$ are developed we provide a signed version of Poisson formula for $R_Δ$. In our proofs we use a topological arguments and topological version of the Parshin reciprocity laws.

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arXiv2013

On the theory of coconvex bodies

If the complement of a closed convex set in a closed convex cone is bounded, then this complement minus the apex of the cone is called a coconvex set. Coconvex sets appear in singularity theory (they are closely related to Newton diagrams) and in commutative algebra. Such invariants of coconvex sets as volumes, mixed volumes, number of integer points, etc., play an important role. This paper aims at extending various results from the theory of convex bodies to the coconvex setting. These include the Aleksandrov-Fenchel inequality and the Ehrhart duality.

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arXiv2013

Aleksandrov-Fenchel inequality for coconvex bodies

We prove a version of the Aleksandrov-Fenchel inequality for mixed volumes of coconvex bodies. This version is motivated by an inequality from commutative algebra relating intersection multiplicities of ideals.

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arXiv2012

Polynomials invertible in k-radicals

A classic result of Ritt describes polynomials invertible in radicals: they are compositions of power polynomials, Chebyshev polynomials and polynomials of degree at most 4. In this paper we prove that a polynomial invertible in radicals and solutions of equations of degree at most k is a composition of power polynomials, Chebyshev polynomials, polynomials of degree at most k and, if k < 15, certain polynomials with exceptional monodromy groups. A description of these exceptional polynomials is given. The proofs rely on classification of monodromy groups of primitive polynomials obtained by Müller based on group-theoretical results of Feit and on previous work on primitive polynomials with exceptional monodromy groups by many authors.

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arXiv2012

Signatures of Branched Coverings

In this paper we deal with branched coverings over the complement to finitely many exceptional points on the Riemann sphere having the property that the local monodromy around each of the branching points is of finite order. To such a covering we assign its \textit{signature}, i.e. the set of its exceptional and branching points together with the orders of local monodromy operators around the branching points. What can be said about the monodromy group of a branched covering if its signature is known? It seems at first that the answer is nothing or next to nothing. Indeed, generically it is so. However there is a (small) list of signatures of \textit{elliptic} and \textit{parabolic} types, for which the monodromy group can be described completely, or at least determined up to an abelian factor. This appendix is devoted to investigation of these signatures. For all these signatures (with one exception) the corresponding monodromy groups turn out to be solvable. Linear differential equations of Fuchs type related to these signatures are solvable in quadratures (in the case of elliptic signatures --- in algebraic functions). A well-known example of this type is provided by Euler differential equations, which can be reduced to linear differential equations with constant coefficients. The algebraic functions related to all (but one) of these signatures are expressible in radicals. A simple example of this kind is provided by the possibility to express the inverse of a Chebyshev polynomial in radicals. Another example of this kind is provided by functions related to division theorems for the argument of elliptic functions. Such functions play a central role in the work [1] of Ritt.

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arXiv2011

Branching Data for Algebraic Functions and Representability by Radicals

The branching data of an algebraic function is a list of orders of local monodromies around branching points. We present branching data that ensure that the algebraic functions having them are representable by radicals. This paper is a review of recent work by the authors and of closely related classical work by Ritt.

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