Any-dimensional Positivstellensätze for symmetric functions

Abstract

Positivstellensätze provide certificates of positivity for polynomials. Extending these certificates to symmetric functions, uniformly across all dimensions, presents structural challenges. For instance, the underlying domain is not semialgebraic. In this paper, we prove two Positivstellensätze for symmetric functions that are uniformly bounded below by some $\epsilon >0$. These are infinite-dimensional analogs of theorems of Pólya and Reznick. The proof relates evaluations of the (truncated) power sum map $(p_2,p_3,\dots )$ to moments of discrete probability measures on the compact interval $[-1,1]$. This yields a characterization of the closure of the orbit space of the infinite symmetric group on the sphere. Finally, we provide an alternative proof of existing Positivstellensätze for normalized symmetric functions.

30 pages. Added Reznick-type Positivstellensätze for non-homogeneous symmetric functions and those involving the first power sum