Positivity Certificates via Integral Representations

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 Gårding'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.

Simple Formula for Integration of Polynomials on a Simplex

“…Define the function h f : R n + → R + , by: z → Γ(1 + n + t) ∆z f (x) dx. Then h f is the Laplace transform of f , or equivalently in the terminology of Kozhasov et al [7], h f is completely monotone 1 (by the Bernstein-Hausdorff-Widder-Choquet theorem; see [7,Theorem 2.5]). Next, let C ⊂ R n be a cone with dual cone…”

“…Interestingly and somehow related, recently Kozhasov et al [7] have considered integration of a "monomial" y α−e (with 0 < α ∈ R n ) with respect to exponential density on the positive orthant. In [7] the density y → i Γ(α i ) y αi−1 i is called the Riesz kernel of the monomial x −α .…”

“…Interestingly and somehow related, recently Kozhasov et al [7] have considered integration of a "monomial" y α−e (with 0 < α ∈ R n ) with respect to exponential density on the positive orthant. In [7] the density y → i Γ(α i ) y αi−1 i is called the Riesz kernel of the monomial x −α . The Riesz kernel offers a certificate of positivity for a function to be completely monotone (a strong positivity property of functions on cones).…”