# Positivity Certificates via Integral Representations

## TL;DR

This paper investigates positivity certificates for functions related to hyperbolic polynomials, establishing conditions for complete monotonicity using integral representations and hypergeometric functions, and proving parts of a conjecture by Scott and Sokal.

## Contribution

It introduces a new approach to certifying complete monotonicity via Riesz kernels and hypergeometric functions, and proves the conjecture for elementary symmetric functions.

## Key findings

- Complete monotonicity holds for sufficiently negative powers of elementary symmetric functions.
- Small negative powers of these polynomials are not completely monotone.
- The Riesz kernel is expressed as a hypergeometric function related to polytope volumes.

## 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.