AI 中文总结
本文证明了具有半平面性质的齐次多项式存在完全单调负幂，并验证了相关正性猜想，给出了幂指数下界及 Riesz 核的严格对数凹性。
AI 中文摘要
Scott 和 Sokal 曾询问是否每个具有半平面性质的齐次多项式都有完全单调的负幂。我们证明确实如此，并证明了 Michałek--Sturmfels--Uhler--Zwiernik 和 Kozhasov--Michałek--Sturmfels 的正性猜想。对于在 $n\ge2$ 个变量中、在其双曲锥上为正的齐次双曲多项式 $p$，每个幂 $p^{-\alpha}$ 在 $\alpha\ge4096n^2$ 时都是完全单调的，这与 $p$ 的次数和系数无关。当锥具有尖闭包时，其 Riesz 核在闭包内部严格对数凹，相对高斯比较误差至多为 $512n^2/\alpha$，并具有显式的 Hessian 界。我们还证明了，完全单调性的正指数只能在 $p$ 是实线性形式的乘积时才能累积到零。特殊的 V'amos 四次曲线提供了一个例子，其任何正整数幂都没有确定的行列式表示。
英文摘要
Scott and Sokal asked whether every homogeneous polynomial with the half-plane property has a completely monotone negative power. We answer this question affirmatively and prove the Riesz-positivity conjecture of Michałek, Sturmfels, Uhler and Zwiernik, restated by Kozhasov, Michałek and Sturmfels: in $n$ variables, every exponent $α\ge4096n^2$ is admissible, independently of the degree and coefficients. For complete hyperbolic polynomials the Riesz density is strictly log-concave, with relative Gaussian error at most $512n^2/α$ and explicit curvature bounds. We characterize admissible exponents by a common spectral Dirichlet law, prove $n\le m+αm(m-1)$ with its equality case, and obtain the sharp degree-dependent gap $0<α<1/(2(m-1))$ whenever the degree-$m$ polynomial has a nonlinear irreducible factor. A nonnegative fourth-order defect of $-\log p$ vanishes at one point precisely for products of positive integer powers of Euclidean Jordan determinants. We classify the corresponding logarithmic Monge-Amp{è}re equation and answer the question of Etingof, Kazhdan and Polishchuk about polynomial multiplicative Legendre transforms within irreducible complete hyperbolic polynomials; the general question has counterexamples, the Clifford quartics of Kogiso and Sato. The classification extends to reducible polynomials under a boundary-visibility hypothesis, and asymptotic common-power formulas for the Riesz densities force exact Jordan formulas.
Comments54 pages; expanded version with spectral Dirichlet representations, sharp lower bounds for admissible exponents, Jordan rigidity, and polynomial multiplicative Legendre transforms; title changed; computational supplement updated
