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 Michałek, Sturmfels, Uhler and Zwiernik, restated by Kozhasov, Michałek and Sturmfels: in n variables, every exponent α≥4096n2 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 512n2/α and explicit curvature bounds. We characterize admissible exponents by a common spectral Dirichlet law, prove n≤ 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.
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BibTeX
@misc{wei2026riesz,
title = {{Riesz kernels of hyperbolic polynomials: positivity, admissible exponents and Jordan rigidity}},
author = {Dongsheng Wei},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31592},
eprint = {2609.31592},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31592},
}
RIS
TY - UNPB
TI - Riesz kernels of hyperbolic polynomials: positivity, admissible exponents and Jordan rigidity
AU - Dongsheng Wei
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.31592
UR - https://arxiv.org/abs/2609.31592
AB - 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≥4096n^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≤ 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.
ER -