  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.
