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Multihomogeneous Nonnegative Polynomials and Sums of Squares

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Abstract

We refine and extend quantitative bounds on the fraction of nonnegative polynomials that are sums of squares to the multihomogeneous case.

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Acknowledgements

I would like to thank Greg Blekherman for useful discussions over e-mail. Ideas developed in Greg Blekherman’s articles had a strong influence on parts of this note. I also would like to thank Petros Valettas and Grigoris Paouris for helpful discussions and splendid hospitality at Athens, College Station and wherever else we were able to meet. While I was writing this note, I was enjoying hospitality of Özgur Kişisel at METU, many thanks go to him. Last but not the least, I would like to thank J. Maurice Rojas for introducing me to quantitative aspects of Hilbert’s \({17}\mathrm{th}\) problem, and for many useful discussions.

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  1. Institut für Mathematik, Technische Universität Berlin, Sekretariat MA 3-2, Straße des 17. Juni 136, 10623, Berlin, Germany

    Alperen A. Ergür

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  1. Alperen A. Ergür
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Correspondence to Alperen A. Ergür.

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Editor in Charge: János Pach

Partially supported by NSF-MCS Grant DMS-0915245, NSF-CAREER Grant DMS-1151711, and Einstein Foundation, Berlin.

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Ergür, A.A. Multihomogeneous Nonnegative Polynomials and Sums of Squares. Discrete Comput Geom 60, 318–344 (2018). https://doi.org/10.1007/s00454-018-0011-3

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  • DOI: https://doi.org/10.1007/s00454-018-0011-3

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