Eventual positivity of Hermitian polynomials and integral operators

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 o...

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Bibliographic Details
Published inarXiv.org
Main Author Tan, Colin
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 04.12.2014
Subjects
Asymptotic series
Hermite polynomials
Integrals
Multiplication
Operators (mathematics)
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Summary: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.
ISSN:2331-8422