Eventual positivity of Hermitian polynomials and integral operators

TL;DR

This paper provides an elementary proof that a certain integral operator associated with Hermitian polynomials is positive-definite, supporting Quillen's result on the positivity of Hermitian polynomials through integral operators.

Contribution

It offers a simplified, elementary proof of the positive-definiteness of the integral operator linked to Hermitian polynomials, avoiding complex asymptotic expansions.

Findings

Elementary proof of positive-definiteness established

Supports Quillen's positivity theorem for Hermitian polynomials

Clarifies the role of integral operators in positivity results

Abstract

Quillen proved that, if a Hermitian bihomogeneous polynomial is strictly positive on the unit sphere, then repeated multiplication of the standard sesquilinear form to this polynomial eventually results in a sum of Hermitian squares. Catlin-D'Angelo and Varolin deduced this positivstellensatz of Quillen from the eventual positive-definiteness of an associated integral operator. Their arguments involve asymptotic expansions of the Bergman kernel. The goal of this article is to give an elementary proof of the positive-definiteness of this integral operator.