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arXiv · 2607.09168

G{\aa}rding's Theorem for Posynomials

Nima Anari
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Abstract

We extend G{\aa}rding'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 $\alpha\pi$ implies $\alpha$-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.

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Nima Anari. 2026-07-10. G{\aa}rding's Theorem for Posynomials. https://arxiv.org/abs/2607.09168

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