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Eventual positivity of Hermitian polynomials and integral operators

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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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  1. Department of Mathematics, National University of Singapore, Block S17, 10, Lower Kent Ridge Road, Singapore, 119076, Singapore

    Colin Tan

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  1. Colin Tan
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Correspondence to Colin Tan.

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Tan, C. Eventual positivity of Hermitian polynomials and integral operators. Chin. Ann. Math. Ser. B 37, 83–94 (2016). https://doi.org/10.1007/s11401-015-0929-1

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  • DOI: https://doi.org/10.1007/s11401-015-0929-1

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