Real Algebraic and Analytic Geometry |

Sums of squares on real algebraic curves.

e-mail:

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

Submission: 2002, September 21.

*Abstract:
Given an affine algebraic variety V over IR with compact set
V(IR) of real points, and a non-negative polynomial function f\in
IR[V] with finitely many real zeros, we establish a local-global
criterion for f to be a sum of squares in IR[V]. We then
specialize to the case where V is a curve. The notion of virtual
compactness is introduced, and it is shown that in the local-global
principle, compactness of V(IR) can be relaxed to virtual
compactness. The irreducible curves are classified on which every
non-negative polynomial is a sum of squares. All results are extended
to the more general framework of preorders. Moreover, applications to
the K-moment problem from analysis are given. In particular,
Schmüdgen's solution of the K-moment problem for compact K is
extended, for dim(K)=1, to the case when K is only virtually compact.*

Mathematics Subject Classification (1991): 14P05,11E25,14H99,14P10,44A60.

Keywords and Phrases: Non-negative polynomials, sums of squares, positivity, preorders, archimedean, real algebraic curves, moment problem.

**Full text**, 30p.:
dvi 170k,
ps.gz 112k,
pdf 355k.

Server Home Page