Gårding's Theorem for Posynomials

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.