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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
66/2019

Positivity Certificates via Integral Representations

Khazhgali Kozhasov, Mateusz Michałek and Bernd Sturmfels

Abstract

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.

Received:
13.08.19
Published:
14.08.19

Related publications

inBook
2022 Repository Open Access
Khazhgali Kozhasov, Mateusz Michałek and Bernd Sturmfels

Positivity certificates via integral representations

In: Facets of algebraic geometry : a collection in honor of William Fulton's 80th birthday ; Volume 2 / Paolo Aluffi... (eds.)
Cambridge : Cambridge University Press, 2022. - pp. 84-114