Gårding's Theorem for Posynomials

2026-07-10

AI Summary

This work extends Gårding’s theorem for the first time to homogeneous positive-coefficient polynomials with arbitrary non-negative real exponents, establishing a precise correspondence between zero-freeness in product domains of the right half-plane and the concavity of degree-normalized root functions. By integrating tools from complex analysis, convex analysis, and the theory of positive polynomials—augmented by AI-assisted derivations—the study uncovers an intrinsic connection between zero-freeness in sectorial regions and fractional-order log-concavity. These findings strengthen theoretical guarantees for fixed-size matchings in combinatorial optimization and for asymmetric determinantal point processes, yielding improved bounds on mixing times and sparsification.

Abstract

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.

Problem

• Gårding's theorem
• posynomials
• zero-freeness
• concavity
• log-concavity

Innovation

• Gårding's theorem
• posynomials
• log-concavity
• zero-freeness
• determinantal point processes