On Hilbert’s 17th problem in low degree

Benoist, Olivier

doi:10.2140/ant.2017.11.929

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On Hilbert's 17th problem in low degree

Olivier Benoist

Vol. 11 (2017), No. 4, 929–959

DOI: 10.2140/ant.2017.11.929

Abstract

Artin solved Hilbert’s 17th problem, proving that a real polynomial in $n$ variables that is positive semidefinite is a sum of squares of rational functions, and Pfister showed that only $2^{n}$ squares are needed.

In this paper, we investigate situations where Pfister’s theorem may be improved. We show that a real polynomial of degree $d$ in $n$ variables that is positive semidefinite is a sum of $2^{n} - 1$ squares of rational functions if $d \leq 2 n - 2$ . If $n$ is even or equal to $3$ or $5$ , this result also holds for $d = 2 n$ .

Keywords

Hilbert's 17th problem, sums of squares, real algebraic geometry, Bloch–Ogus theory

Mathematical Subject Classification 2010

Primary: 11E25

Secondary: 14F20, 14P99

Milestones

Received: 11 July 2016

Revised: 5 January 2017

Accepted: 3 February 2017

Published: 18 June 2017

Authors

Olivier Benoist
Institut de Recherche Mathématique Avancée UMR 7501, Université de Strasbourg et CNRS 7 rue René Descartes 67000 Strasbourg France

Vol. 11, No. 4, 2017