Kozhasov Khazhgali: On complete monotonicity of inverse powers of elementary symmetric polynomials

Abstract

Sufficiently high inverse powers \(p^a\), \(a<0\), of some real stable polynomials \(p\) (among which are basis generating polynomials of certain classes of matroids) turn out to be completely monotone, that is, the coefficients of the Taylor expansion of \(y \mapsto p(x-y)^a\) are nonnegative for any \(x\) in the positive orthant. I will discuss this phenomenon for the class of elementary symmetric polynomials. The talk is based on a joint work with M. Mikhalek and B. Sturmfels.