August Tsikh

Multidimensional Residues and Their Applications

Preliminary information Residues associated with mappings

Residue currents of the Bochner-Martinelli type

f\colon \mathbf C^n\rightarrow \mathbf C^n

Algebraic Equations and Hypergeometric Series

(local residues) Residues associated with mappings

Singularities of hypergeometric functions in several variables

f\colon \mathbf C^n\rightarrow \mathbf C^p

Residual kernels with singularities on coordinate planes

(residual currents and principal values) Applications to function theory and algebraic geometry Applications to the calculation of double series and integrals.

Amoebas of Complex Hypersurfaces in Statistical Thermodynamics

Our objective is to construct residue currents from Bochner-Martinelli type kernels; the computations hold in the non complete intersection case and provide a new and more direct approach of the residue of Coleff-Herrera in the complete intersection case; computations involve crucial relations with toroidal varieties and multivariate integrals of the Mellin-Barnes type.

Laurent Determinants and Arrangements of Hyperplane Amoebas

We study the solutions of a general n th order algebraic equation represented by multidimensional hypergeometric series. We provide a detailed description of the domains of convergence of these series in terms of the amoeba and the Horn-Kapranov uniformization of the corresponding discriminant. From a geometric viewpoint this amounts to describing the maximal Reinhardt domains in the complement of the discriminant locus.

Residue integrals and their Mellin transforms

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.

Multidimensional versions of Poincaré's theorem for difference equations

A finite collection of planes {Ev} in ℂd is called an atomic family if the top de Rham cohomology group of its complement is generated by a single element. A closed differential form generating this group is called a residual kernel for the atomic family. We construct new residual kernels in the case when Ev are coordinate planes such that the complement ℂd/∪ Ev admits a toric action with the orbit space being homeomorphic to a compact projective toric variety. They generalize the well-known Bochner-Martinelli and Sorani differential forms. The kernels obtained are used to establish a new formula of integral representations for functions holomorphic in Reinhardt polyhedra.

SINGULARITIES OF NONCONFLUENT HYPERGEOMETRIC FUNCTIONS IN SEVERAL VARIABLES

The amoeba of a complex hypersurface is its image under the logarithmic projection. A number of properties of algebraic hypersurface amoebas are carried over to the case of transcendental hypersurfaces. We demonstrate the potential that amoebas can bring into statistical physics by considering the problem of energy distribution in a quantum thermodynamic ensemble. The spectrum