manuscripta mathematica

, Volume 119, Issue 4, pp 395–410

Sums of squares on real algebraic surfaces

Article

DOI: 10.1007/s00229-006-0630-5

Cite this article as:
Scheiderer, C. manuscripta math. (2006) 119: 395. doi:10.1007/s00229-006-0630-5

Abstract

Consider real polynomials g1, . . . , gr in n variables, and assume that the subset K = {g1≥0, . . . , gr≥0} of ℝn is compact. We show that a polynomial f has a representation

https://static-content.springer.com/image/art%3A10.1007%2Fs00229-006-0630-5/MediaObjects/s00229-006-0630-5flb1.gif

in which the se are sums of squares, if and only if the same is true in every localization of the polynomial ring by a maximal ideal. We apply this result to provide large and concrete families of cases in which dim (K) = 2 and every polynomial f with f|K≥0 has a representation (*). Before, it was not known whether a single such example exists. Further geometric and arithmetic applications are given.

Mathematics Subject Classification (2000)

Primary 14P05secondary 11E25

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Fachbereich Mathematik und StatistikUniversität KonstanzKonstanzGermany