Eventual positivity of Hermitian polynomials and integral operators

Colin Tan

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arXiv · 1412.1570

Eventual positivity of Hermitian polynomials and integral operators

Colin Tan

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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.

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Colin Tan. 2014-12-04. Eventual positivity of Hermitian polynomials and integral operators. https://arxiv.org/abs/1412.1570

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