Timur Mradovich Sadykov

Plekhanov Russian Academy of Economics · Mathematics and Computer Science

With any integer convex polytope \(P\subset {\mathbb {R}}^n\) we associate a multivariate hypergeometric polynomial whose set of exponents is \({\mathbb {Z}}^{n}\cap P.\) This polynomial is defined uniquely up to a constant multiple and satisfies a holonomic system of partial differential equations of Horn’s type. We prove that under certain nondeg...

The book provides an introduction into the modern theory of hypergeometric and algebraic functions in several complex variables. It also gives a comprehensive survey of selected recent results in the field.

We introduce the notion of the analytic complexity of a cluster tree and develop an algorithm to compute it.

We revisit the classical problem of construction of a fundamental system of solutions to a linear ODE whose elements remain analytic and linearly independent for all values of the roots of the characteristic polynomial.

We present a set of deterministic algorithms for Russian inflection and automated text synthesis. These algorithms are implemented in a publicly available web-service www.passare.ru. This service provides functions for inflection of single words, word matching and synthesis of grammatically correct Russian text. Selected code and datasets are avail...

This is a Mathematica package for construction, investigation, and inversion of polynomial and certain more general analytic mappings with unit determinant of the Jacobian matrix. The package comes with a library of datasets obtained by means of computational experiments.

This is the poster of the international workshop on Computer Algebra in Scientific Computing to be held at Gebze Technical University on August 22-26, 2022.

Let $x=(x_1,\ldots,x_n)\in {\rm \bf C}^n$ be a vector of complex variables, denote by $A=(a_{jk})$ a square matrix of size $n\geq 2,$ and let $\varphi\in\mathcal{O}(\Omega)$ be an analytic function defined in a nonempty domain $\Omega\subset {\rm \bf C}.$ We investigate the family of mappings $$ f=(f_1,\ldots,f_n):{\rm \bf C}^n\rightarrow {\rm \bf...

This book constitutes the proceedings of the 23rd International Workshop on Computer Algebra in Scientific Computing, CASC 2021, held in Sochi, Russia, in September 2021. The 24 full papers presented together with 1 invited talk were carefully reviewed and selected from 40 submissions. The papers cover theoretical computer algebra and its applicati...

This book constitutes the refereed proceedings of the 22nd International Workshop on Computer Algebra in Scientific Computing, CASC 2020, held in Linz, Austria, in September 2020. The conference was held virtually due to the COVID-19 pandemic. The 34 full papers presented together with 2 invited talks were carefully reviewed and selected from 41 su...

The international conference on Computer Algebra in Scientific Computing will be held at Johannes Kepler University in Linz, Austria on September 14-18, 2020. For details, please visit http://www.casc-conference.org/

This is a collection of highly complex amoebas of optimal bivariate polynomials. It represents a technical spin-off of the research recently published in Mathematische Zeitschrift. These amoebas are currently on the border of the set of those being efficiently computable by means of our methods. All of the featured polynomials are dense and suppor...

Algorithms for computing rational generating functions of solutions of one-dimensional difference equations are well-known and easy to implement. We propose an algorithm for computing rational generating functions of solutions of two-dimensional difference equations in terms of initial data of the corresponding initial value problems. The crucial p...

With our new HP workstation, computation of highly complex amoebas is now within reach.

The hypergeometric polynomial whose amoeba is depicted in the figure Giant amoeba.png

This book constitutes the refereed proceedings of the 21st International Workshop on Computer Algebra in Scientific Computing, CASC 2019, held in Moscow, Russia, in August 2019. The 28 full papers presented together with 2 invited talks were carefully reviewed and selected from 44 submissions. They deal with cutting-edge research in all major disci...

The second International conference is organized jointly by Dorodnicyn Computing Center of Federal Research Center "Computer Science and Control" of RAS and Plekhanov Russian University of Economics with a support of the Russian Foundation for Basic Research (project 17-01-20398 17). The first edition of the event [1] (http://www.ccas.ru/ca/confere...

We consider computational problems of the theory of hypergeometric functions in several complex variables: computation of the holonomic rank of a hypergeometric system of partial differential equations, computing the defining polynomial of the singular hypersurface of such a system and finding its monomial solutions. The presented algorithms have b...

Given a complex algebraic hypersurface~$H$, we introduce a polyhedral complex which is a subset of the Newton polytope of the defining polynomial for~$H$ and enjoys the key topological and combinatorial properties of the amoeba of~$H.$ We provide an explicit formula for this polyhedral complex in the case when the spine of the amoeba is dual to a t...

This is a Mathematica package for the analysis of holonomic systems of hypergeometric partial differential equations with two variables. It allows one to compute the holonomic rank, the defining polynomial for the singular hypersurface, monomial and polynomial solutions to the system in question and generating solutions for atomic hypergeometric sy...

Hypergeometric functions of several variables resemble functions of finite analytic complexity in the sense that the elements of both classes satisfy certain canonical overdetermined systems of partial differential equations. Otherwise these two sets of functions are very different. We investigate the relation between the two classes of functions a...

We present a deterministic algorithm for Russian inflection. This algorithm is implemented in a publicly available web-service www.passare.ru which provides functions for inflection of single words, word matching and synthesis of grammatically correct Russian text. The inflectional functions have been tested against the annotated corpus of Russian...

We propose an algorithm for evaluation of rational generating functions for solutions of the Cauchy problems for two-dimensional difference equations with constant coefficients. The coefficients of onedimensional difference equations and the initial data are used to solve the corresponding Cauchy problems. The algorithm is implemented in the Maple...

We present algorithms for computation and visualization of polynomial amoebas, their contours, compactified amoebas and sections of three-dimensional amoebas by two-dimensional planes. We also provide a method and an algorithm for the computation of polynomials whose amoebas exhibit the most complicated topology among all polynomials with a fixed N...

Inflection of Russian nouns, verbs, and adverbs, declension of adjectives, full algorithmic coverage of Russian grammar, automated text synthesis are among the services offered by the website www.passare.ru.

We present algorithms for computation and visualization of amoebas, their contours, compactified amoebas and sections of three-dimensional amoebas by two-dimensional planes. We also provide method and an algorithm for the computation of~polynomials whose amoebas exhibit the most complicated topology among all polynomials with a fixed Newton polytop...

The frequencies of human blood genotypes in the ABO and Rh systems differ between populations. Moreover, in a given population, these frequencies typically evolve over time. The possible reasons for the existing and expected differences in these frequencies (such as disease, random genetic drift, founder effects, differences in fitness between the...

With any integer convex polytope P in the real space we associate a multivariate hypergeometric polynomial whose set of exponents is the set of integer points in P. This polynomial is defined uniquely up to a constant multiple and satisfies a holonomic system of partial differential equations of Horn's type. We prove that the zero locus of any such...

With any integer convex polytope P in the real space we associate a multivariate hypergeometric polynomial whose set of exponents is the set of integer points in P. This polynomial is defined uniquely up to a constant multiple and satisfies a holonomic system of partial differential equations of Horn's type. We prove that the zero locus of any such...

We define the notion of the analytic complexity of a binary cluster tree and investigate its properties.

We give a complete closed form description of the evolution of human blood genotypes frequencies (in the ABO and Rh classification) after any (finite or infinite) number of generations and for any initial distribution.

We investigate branching of solutions to holonomic bivariate hypergeometric systems of Horn type. Special attention is paid to the invariant subspace of Puiseux polynomial solutions. We mainly study (1) Horn systems defined by simplicial configurations, (2) Horn systems whose Ore-Sato polygon is either a zonotope or a Minkowski sum of a triangle an...

The paper deals with the notion of analytic complexity introduced by V.K. Beloshapka. We give an algorithm which allows one to check whether a bivariate analytic function belongs to the second class of analytic complexity. We also provide estimates for the analytic complexity of classical discriminants and introduce the notion of analytic complexit...

We give a computationally efficient method for constructing the linear differential operator with polynomial coefficients whose space of holomorphic solutions is spanned by all the branches of a function defined by a generic algebraic curve. The proposed method does not require solving the algebraic equation and can be applied in the case when its...

A discrete version of the classical Riemann-Hilbert problem is stated and solved. In particular, a Riemann-Hilbert problem is associated with every dessin d’enfants. It is shown how to compute the solution for a dessin that is a tree. This amounts to finding a Fuchsian differential equation satisfied by the local inverses of a Shabat polynomial. A...

The study is carried out with a purpose to solve the closely related problem of describing hypergeometric systems whose holomorphic solution space is the direct sum of one-dimensional invariant subspaces. The Zonotype is a polyhedron being the Minkowski sum of intervals, where a theorem has been proposed to get the results. The Hypergeometric syste...

We prove that any simplicial or parallelepipedal hypergeometric configuration admits a Puiseux polynomial basis in its solution space for suitable values of its parameters.

The algebraicity of solution space of the Mellin system is discussed. A basis is constructed in the solution space to prove that the monodromy of the Mellin system is reducible. Solutions to this system are derived from closed-form formulas. The formula for the number of linearly independent algebraic solutions to the Mellin system is proved with t...

Local holomorphic solutions z=z(a) to a univariate sparse polynomial equation p(z) =0, in terms of its vector of complex coefficients a, are classically known to satisfy holonomic systems of linear partial differential equations with polynomial coefficients. In this paper we investigate one of such systems of differential equations which was introd...

We state and solve a discrete version of the classical Riemann-Hilbert problem. In particular, we associate a Riemann-Hilbert problem to every dessin d'enfants. We show how to compute the solution for a dessin that is a tree. This amounts to finding a Fuchsian differential equation satisfied by the local inverses of a Shabat polynomial. We produce...

The paper deals with singularities of nonconfluent hypergeometric functions in several variables. Typically such a function is a multi-valued analytic function with singularities along an algebraic hypersurface. We describe such hypersurfaces in terms of amoebas and the Newton polytopes of their defining polynomials. In particular, we show that all...

We undertake the study of bivariate Horn systems for generic parameters. We prove that these hypergeometric systems are holonomic, and we provide an explicit formula for their holonomic rank as well as bases of their spaces of complex holomorphic solutions. We also obtain analogous results for the generalized hypergeometric systems arising from lat...

The paper deals with the Horn system of hypergeometric differential equations. We consider the associated D-module and construct an explicit basis in the space of holomorphic solutions to the Horn system under some assumptions on its parameters.

Typically a hypergeometric function is a multi-valued analytic function with algebraic singularities. In this paper we give a complete description of the Newton polytope of the polynomial whose zero set naturally contains the singular locus of a nonconfluent double hypergeometric series. We show in particular that the Hadamard multiplication of suc...

This thesis deals with systems of partial differential equations of hypergeometric type and singularities of hypergeometric functions. By definition hypergeometric functions are (multi-valued) analytic solutions to the so-called Horn system of equations.The main purpose of the thesis is to systematically investigate the Horn system of equations and...

We study the Horn system of partial differential equations which generalizes the ordinary hypergeometric differential equation to the case of several complex variables. Using one of the variants of the Laplace transform we associate a system of difference equations of the first order with polynomial coefficients to the original system of partial di...