# Sums of squares on real algebraic surfaces

- First Online:

- Received:
- Revised:

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 *g*_{1}, . . . , *g*_{r} in *n* variables, and assume that the subset *K* = {*g*_{1}≥0, . . . , *g*_{r}≥0} of ℝ^{n} is compact. We show that a polynomial *f* has a representation

in which the *s*_{e} 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.