From a formal perspective tropical geometry can be viewed as a branch of geometry
manipulating with certain piecewise-linear objects that take over the role of classical algebraic
varieties. This talk outlines some basic notions of this area and surveys some of its applications
for the problems in classical (real and complex) geometry.

http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_40.pdf

Tropical geometry and its applications

These notes outline some basic notions of Tropical Geometry and survey some of its applications for problems in classical (real and complex) geometry. To appear in the Proceedings of the Madrid ICM.

Tropical Geometry and its applications

An explicit lattice point realization is provided for the primary components of an arbitrary binomial ideal in characteristic zero This decomposition is derived from a characteristic-free combinatorial description of certain primary components of binomial ideals in affine semigroup rings, namely those that are associated to faces of the semigroup These results are intimately connected to hypergeometric differential equations in several variables

/pdf/combinatorics-of-binomial-primary-decomposition-a7gwuiiemh.pdf

Combinatorics of binomial primary decomposition

The simplest partition function, associated with homogeneous symmetric forms S of degree r in n variables, is integral discriminant Jn|r(S) = ∫e−S(x1,...,xn)dx1...dxn. Actually, S-dependence remains the same if e−S in the integrand is substituted by arbitrary function f(S), i.e. integral discriminant is a characteristic of the form S itself, and not of the averaging procedure. The aim of the present paper is to calculate Jn|r in a number of non-Gaussian cases. Using Ward identities — linear differential equations, satisfied by integral discriminants — we calculate J2|3,J2|4,J2|5 and J3|3. In all these examples, integral discriminant appears to be a generalized hypergeometric function. It depends on several SL(n) invariants of S, with essential singularities controlled by the ordinary algebraic discriminant of S.

http://iopscience.iop.org/article/10.1088/1126-6708/2009/12/002/pdf

Introduction to integral discriminants

http://mate.dm.uba.ar/%7Ealidick/papers/aliberc.pdf

Elimination Theory in Codimension 2

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.

/pdf/singularities-of-hypergeometric-functions-in-several-444x2cgbqk.pdf

Singularities of hypergeometric functions in several variables

Bivariate hypergeometric D-modules

This thesis deals with hypergeometric functions in several complex variables and systems of partial differential equations of hypergeometric type. One of the main objects of study in the thesis is the so-called Horn system of equations: xiPi(θ)y(x) = Qi(θ)y(x), i = 1, ..., n.Here x ∈ℂn, θ = (θ1, ..., θn),θi = xi ∂/∂xi , Pi and Qi are nonzero polynomials. By definition hypergeometric functions are (multi-valued) analytic solutions to this system of equations. The main purpose of the thesis is to systematically investigate the Horn system of equations and properties of its solutions.To construct solutions to the Horn system we use one of the variants of the Laplace transform which leads to a system of linear difference equations with polynomial coefficients. Solving this system we represent a solution to the Horn system in the form of an iterated Puiseux series.We give an explicit formula for the dimension of the space of analytic solutions to the Horn system at a generic point under some assumptions on its parameters. The proof is based on the study of the module over the Weyl algebra of linear differential operators with polynomial coefficients associated with the Horn system. Combining this formula with the theorem which allows one to represent a solution to the Horn system in the form of an iterated Puiseux series, we obtain a basis in the space of analytic solutions to this system of equations.Another object of study in the thesis is the singular set of a nonconuent hypergeometric function in several variables. Typically such a function is a multi-valued analytic function with singularities along an algebraic hypersurface. We give a description of such hypersurfaces in terms of the Newton polytopes of their defining polynomials. In particular we obtain a geometric description of the zero set of the discriminant of a general algebraic equation.In the case of two variables one can say much more about singularities of nonconuent hypergeometric functions. We give a complete description of the Newton polytope of the polynomial whose zero set naturally contains the singular locus of a nonconuent double hypergeometric series. We show in particular that the Hadamard multiplication of such series corresponds to the Minkowski sum of the Newton polytopes of polynomials whose zero loci contain the singularities of the factors.

Nonconfluent hypergeometric functions in several variables and their singularities

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.

Timur Sadykov

Papers

Singularities of hypergeometric functions in several variables

Bivariate hypergeometric D-modules

Nonconfluent hypergeometric functions in several variables and their singularities

Bases in the solution space of the Mellin system

Algebraicity of solutions to the Mellin system and its monodromy