{"ID":6497788,"CreatedAt":"2026-07-13T01:19:40.13847098Z","UpdatedAt":"2026-07-14T01:36:59.12045529Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.09168","arxiv_id":"2607.09168","title":"Gårding's Theorem for Posynomials","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.","short_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-concavit...","url_abs":"https://arxiv.org/abs/2607.09168","url_pdf":"https://arxiv.org/pdf/2607.09168v1","authors":"[\"Nima Anari\"]","published":"2026-07-10T07:52:56Z","proceeding":"cs.DS","tasks":"[\"cs.DS\",\"math.CO\",\"math.PR\"]","methods":"[]","has_code":false}