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MiS Preprint
66/2019
Positivity Certificates via Integral Representations
Khazhgali Kozhasov, Mateusz Michałek and Bernd Sturmfels
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.
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
