Riesz kernels of hyperbolic polynomials: positivity, admissible exponents and Jordan rigidity

Abstract

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 Micha{\l}ek, Sturmfels, Uhler and Zwiernik, restated by Kozhasov, Micha{\l}ek and Sturmfels: in $n$ variables, every exponent $\alpha\ge4096n^2$ 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 $512n^2/\alpha$ and explicit curvature bounds. We characterize admissible exponents by a common spectral Dirichlet law, prove $n\le m+\alpha m(m-1)$ with its equality case, and obtain the sharp degree-dependent gap $0<\alpha<1/(2(m-1))$ 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{\`e}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.