Discrete & Computational Geometry

, Volume 60, Issue 2, pp 318–344 | Cite as

Multihomogeneous Nonnegative Polynomials and Sums of Squares

  • Alperen A. ErgürEmail author


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


Multihomogeneous forms Sums of squares Isotropic measures Hilbert’s 17th problem 

Mathematics Subject Classification

Primary 52A38 Secondary 90C22 



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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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany

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