AI 中文总结
研究将加丁定理扩展到齐次正多项式，通过证明在特定区域无零点时度归一化根的凹性及相关性质，加强了固定大小匹配和非对称行列式点过程的混合与稀疏化保证，借助人工智能辅助得出此结果。
AI 中文摘要
我们将加丁定理扩展到齐次正多项式：若具有任意非负实指数的单项式的有限正和在右半平面的乘积上无零点，则其度归一化根是凹的。因此，在孔径为\(\alpha\pi\)的扇形区域内无零点意味着\(\alpha\) - 分数对数凹性。这加强了固定大小匹配和非对称行列式点过程的一般混合和域稀疏化保证。该结果是在作者发起并检查的人工智能辅助交互中得出的，Codex还协助了稿件的组装和排版。
英文摘要
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.
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