Gårding's Theorem for Posynomials
Provides a theoretical result that sharpens existing guarantees for specific combinatorial and probabilistic models, but is incremental in nature.
The paper extends Gårding's theorem to homogeneous posynomials, showing that zero-freeness on a product of right half-planes implies concavity of the degree-normalized root, and zero-freeness in a sector implies α-fractional log-concavity. This sharpens guarantees for mixing and domain-sparsification in matchings and determinantal point processes.
We extend Gårding's theorem to homogeneous posynomials: if a finite positive sum of monomials with arbitrary nonnegative real exponents is zero-free on a product of right half-planes, then its degree-normalized root is concave. Consequently, zero-freeness in a sector of aperture $απ$ implies $α$-fractional log-concavity. This sharpens generic mixing and domain-sparsification guarantees for fixed-size matchings and nonsymmetric determinantal point processes. The result was developed in an AI-assisted interaction initiated and checked by the author; Codex also assisted with assembling and typesetting the manuscript.