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Positive Semidefinite Analytic Functions on Real Analytic Surfaces

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Abstract

Let \(X\subset {\mathbb {R}}^n\) be a (global) real analytic surface. Then every positive semidefinite meromorphic function on X is a sum of 10 squares of meromorphic functions on X. As a consequence, we provide a real Nullstellensatz for (global) real analytic surfaces.

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Acknowledgements

The author is very grateful to S. Schramm for a careful reading of the final version and for the suggestions to refine its redaction.

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  1. Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040, Madrid, Spain

    José F. Fernando

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Author supported by Spanish STRANO MTM2017-82105-P and Grupos UCM 910444.

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Fernando, J.F. Positive Semidefinite Analytic Functions on Real Analytic Surfaces. J Geom Anal 31, 12375–12410 (2021). https://doi.org/10.1007/s12220-021-00721-1

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