Timur M. Sadykov

Singularities of hypergeometric functions in several variables

This paper deals with singularities of nonconfluent hypergeometric functions in several complex variables. Typically such a function is a multi-valued analytic function with singularities along an algebraic hypersurface. We describe such hypersurfaces in terms of the amoebas and the Newton polytopes of their defining polynomials. In particular, we show that the amoebas of classical discriminantal hypersurfaces are solid, that is, they possess the minimal number of complement components.

Bases in the solution space of the Mellin system

We consider algebraic functions satisfying equations of the following form: (1) Here , , and is a function of the complex variables . Solutions of such algebraic equations are known to satisfy holonomic systems of linear differential equations with polynomial coefficients. In this paper we investigate one such system, which was introduced by Mellin. The holonomic rank of this system of equations and the dimension of the linear space of its algebraic solutions are computed. An explicit base in the solution space of the Mellin system is constructed in terms of roots of (1) and their logarithms. The monodromy of the Mellin system is shown to be always reducible and several results on the factorization of the Mellin operator in the one-variable case are presented. Bibliography: 18 titles.

Maximally reducible monodromy of bivariate hypergeometric systems

We investigate branching of solutions to holonomic bivariate hypergeometric systems of Horns type. Special attention is paid to the invariant subspace of Puiseux polynomial solutions. We mainly study Horn systems defined by simplicial configurations and Horn systems whose Ore-Sato polygons are either zonotopes or Minkowski sums of a triangle and segments proportional to its sides. We prove a necessary and sufficient condition for the monodromy representation to be maximally reducible, that is, for the space of holomorphic solutions to split into the direct sum of one-dimensional invariant subspaces.

Polynomial dynamics of human blood genotypes frequencies

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 various blood groups etc.) are the focus of intensive research. To understand the effects of historical and evolutionary influences on the blood genotypes frequencies, it is important to know how these frequencies behave if no influences at all are present. Under this assumption the dynamics of the blood genotypes frequencies is described by a polynomial dynamical system defined by a family of quadratic forms on the 17-dimensional projective space. To describe the dynamics of such a polynomial map is a task of substantial computational complexity.We give a complete analytic description of the evolutionary trajectory of an arbitrary distribution of human blood variations frequencies with respect to the clinically most important ABO and RhD antigens. We also show that the attracting algebraic manifold of the polynomial dynamical system in question is defined by a binomial ideal.

Computational Problems of Multivariate Hypergeometric Theory

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 been implemented in the computer algebra system MATHEMATICA.