Abstract:
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.