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Sums of three squares and Noether–Lefschetz loci

Part of: Families, fibrations Special varieties Forms and linear algebraic groups

Published online by Cambridge University Press:  03 April 2018

Olivier Benoist
Olivier Benoist*
Affiliation:
Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67000 Strasbourg, France email olivier.benoist@unistra.fr
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Abstract

We show that the set of real polynomials in two variables that are sums of three squares of rational functions is dense in the set of those that are positive semidefinite. We also prove that the set of real surfaces in $\mathbb{P}^{3}$ whose function field has level 2 is dense in the set of those that have no real points.

Keywords

sums of squaresHodge theoryreal algebraic geometryvariations of Hodge structures

MSC classification

Primary: 11E25: Sums of squares and representations by other particular quadratic forms
Secondary: 14P99: None of the above, but in this section 14D07: Variation of Hodge structures 14M12: Determinantal varieties
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 5 , May 2018 , pp. 1048 - 1065
Copyright
© The Author 2018 

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