Book chapter · Cambridge University Press eBooks · March 13, 2022
Positivity Certificates via Integral Representations
Khazhgali Kozhasov, Mateusz Michałek, 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.
More by Khazhgali Kozhasov
- ON THE DISTANCE TO DISCRIMINANTS AND OSCULATING VARIETIESarXiv · 2026
- Stubborn PolynomialsarXiv · 2026
- On odd powers of nonnegative polynomials that are not sums of squaresForum of Mathematics Sigma · 2026 · Citations: 1
- On the minimal algebraic complexity of the rank-one approximation problem for general inner productsMathematics of Computation · 2025 · Citations: 1