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On the bad points of positive semidefinite polynomials

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Abstract

A bad point of a positive semidefinite real polynomial f is a point at which a pole appears in all expressions of f as a sum of squares of rational functions. We show that quartic polynomials in three variables never have bad points. We give examples of positive semidefinite polynomials with a bad point at the origin, that are nevertheless sums of squares of formal power series, answering a question of Brumfiel. We also give an example of a positive semidefinite polynomial in three variables with a complex bad point that is not real, answering a question of Scheiderer.

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Acknowledgements

I thank Charles N. Delzell and Claus Scheiderer for having made [10] available to me, Karim Johannes Becher for useful comments, and the referee for their careful work.

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  1. Département de Mathématiques et Applications, École normale supérieure, CNRS, 45 Rue d’Ulm, 75230, Paris Cedex 05, France

    Olivier Benoist

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  1. Olivier Benoist
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Correspondence to Olivier Benoist.

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Dedicated to Olivier Debarre.

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Benoist, O. On the bad points of positive semidefinite polynomials. Math. Z. 300, 3383–3403 (2022). https://doi.org/10.1007/s00209-021-02804-9

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  • DOI: https://doi.org/10.1007/s00209-021-02804-9

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