Hypergeometric polynomials are optimal

TL;DR

This paper introduces a class of multivariate hypergeometric polynomials associated with convex polytopes, proving their zero loci are optimal in a specific mathematical sense, advancing understanding of their geometric and algebraic properties.

Contribution

It establishes a novel connection between hypergeometric polynomials and convex polytopes, proving the optimality of their zero loci, which was previously unknown.

Findings

Zero locus of hypergeometric polynomials is optimal

Polynomials satisfy a Horn-type holonomic system

Unique up to a constant multiple

Abstract

With any integer convex polytope $P\subset\midR^n$ we associate a multivariate hypergeometric polynomial whose set of exponents is $\midZ^{n}\cap P.$ 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 the zero locus of any such polynomial is optimal in the sense of Forsberg-Passare-Tsikh.