Real Algebraic and Analytic Geometry |

Sums of squares on real algebraic surfaces.

e-mail:

homepage: http://www.math.uni-konstanz.de/~scheider/index.html

Submission: 2004, August 28.

*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
f = \sum_{e\in\{0,1\}^r} s_e.g_1^{e_1}...g_r^{e_r} (*)
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
.*

Mathematics Subject Classification (2000): 14P05, 11E25, 26D05.

Keywords and Phrases: Non-negative polynomials, polynomial inequalities, sums of squares, preorderings, archimedean, real algebraic geometry, real algebraic surfaces, Hilbert 17th problem.

**Full text**, 13p.:
dvi 74k,
ps.gz 191k,
pdf 291k.

Server Home Page