Extremal properties of multivariate hypergeometric polynomials

With any integer convex polytope $P\subset\mathbb R^n$ we associate a multivariate hypergeometric polynomial whose set of exponents is $\mathbb Z^{n}\cap P$. This polynomial is defined uniquely up to a constant multiple and satisfies a holonomic system of partial differential equations of Horn's type. Special instances include numerous families of orthogonal polynomials in one and several variables. In the talk, we will discuss several extremal properties of multivariate polynomials defined in this way. In particular, we prove that the zero locus of any such polynomial is optimal in the sense of Forsberg–Passare–Tsikh.