Vol. 11, No. 4, 2017

Download this article
Download this article For screen
For printing
Recent Issues

Volume 11
Issue 6, 1243–1488
Issue 5, 1009–1241
Issue 4, 767–1007
Issue 3, 505–765
Issue 2, 253–503
Issue 1, 1–252

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
Cover
Editorial Board
Editors' Addresses
Editors' Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Subscriptions
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
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