Hypergeometric polynomials are optimal

Abstract

With any integer convex polytope P ⊂ R n we associate a multivariate hypergeometric polynomial whose set of exponents is Z n ∩ 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 under certain nondegeneracy conditions the zero locus of any such polynomial is optimal in the sense of [7], i.e., that the topology of its amoeba [11] is as complex as it could possibly be. Using this, we derive optimal properties of several classical families of multivariate hypergeometric polynomials.

Parameterizing and inverting analytic mappings with unit Jacobian

“…We prove any such mapping to be polynomially invertible and provide an explicit recursive formula for its inverse (see Theorem 6.8). Being able to explicitly invert a polynomial mapping allows one to compute complex amoebas of multivariate polynomials with high precision and to investigate their geometric properties [1]. All of the polynomial mappings constructed in the paper are polynomially invertible.…”