Vol. 11, No. 4, 2017

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On Hilbert's 17th problem in low degree

Olivier Benoist

Vol. 11 (2017), No. 4, 929–959
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 2n 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 2n 1 squares of rational functions if d 2n 2. If n is even or equal to 3 or 5, this result also holds for d = 2n.

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