COMPLEX VARIABLES ANALOGUES OF HILBERT'S SEVENTEENTH PROBLEM

Abstract

We prove several theorems concerning the representation of Hermitian symmetric polynomials as quotients of squared norms of holomorphic polynomial mappings, thus providing complex variables analogues of Hilbert's seventeenth problem. We consider the space of Hermitian symmetric polynomials R on Cn of degree at most d in z, with the Euclidean topology on the space of coefficients. We compare the collections of nonnegative polynomials 𝒫d and quotients of squared norms 𝒬d. We prove, for d ≥ 2, that 𝒬d strictly contains the interior of 𝒫d and is strictly contained in 𝒫d. We provide a tractable precise description of 𝒬d in one dimension. We also give a necessary and sufficient condition in general in terms of F and G in the holomorphic decomposition R = ||F||2 - ||G||2. We provide many surprising examples and counterexamples and briefly discuss some applications.