# Veranstaltung

Polynomial instances of hypergeometric functions in one and several variables are very diverse. They comprise the classical Chebyshev polynomials of the first and the second kind, the Gegenbauer, Hermite, Jacobi, Laguerre and Legendre polynomials as well as their numerous multivariate analogues.

Despite the diversity of families of hypergeometric polynomials, most of them share the following key properties that justify the usage of the term "hypergeometric":

1. The polynomials are dense (possibly after a suitable monomial change of variables).

2. The coefficients of a hypergeometric polynomial are related through some recursion with polynomial coefficients.

3. For univariate polynomials, there is typically a single representative (up to a suitable normalization) of a given degree within a family of hypergeometric polynomials.

4. All polynomials in the family satisfy a differential equation of a fixed order with polynomial coefficients (or a system of such equations) whose parameters encode the degree of a polynomial.

5. In the case of one dimension, the absolute values of the roots of a classical hyper-geometric polynomial are all different (possibly after a suitable monomial change of variables).

6. Many of hypergeometric polynomials enjoy various extremal properties.

In the talk we will introduce a definition of a multivariate hypergeometric polynomial in several complex variables that is coherent with the properties 1-6 listed above. Namely, with any integer convex polytope $P$ we associate a multivariate hypergeometric polynomial whose set of exponents is $P$.

The hypergeometric polynomial associated with the polytope $P$ 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 non-degeneracy conditions the zero locus of any such polynomial is optimal in the sense of Forsberg-Passare-Tsikh. Generally speaking, this means that the topology of the amoeba of such a polynomial is as complicated as it could possibly be. This property is the multivariate counterpart of the property of having different absolute values of the roots for a polynomial in a single variable.