2019/08/12 by Khazhgali Kozhasov, Kozhasov, Khazhgali, Mateusz Michałek +3 · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #Analytic and geometric function theory #Combinatorics (math.CO) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical functions and polynomials #Optimization and Control (math.OC) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1908.04191
openalex publication_date 2019/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Complete monotonicity is a strong positivity property for real-valued functions on convex cones. It is certified by the kernel of the inverse Laplace transform. We study this for negative powers of hyperbolic polynomials. Here the certificate is the Riesz kernel in Garding's integral representation. The Riesz kernel is a hypergeometric function in the coefficients of the given polynomial. For monomials in linear forms, it is a Gel'fand-Aomoto hypergeometric function, related to volumes of polytopes. We establish complete monotonicity for sufficiently negative powers of elementary symmetric functions. We also show that small negative powers of these polynomials are not completely monotone, proving one direction of a conjecture by Scott and Sokal.