 
Summary: POSITIVE POLYNOMIALS ON PROJECTIVE LIMITS
OF REAL ALGEBRAIC VARIETIES
SALMA KUHLMANN AND MIHAI PUTINAR
Abstract. We reveal some key geomteric aspects related to non
convex optimization of sparse polynomials. The main result, a
Positivstellensatz on the fibre product of real algebraic, affine va
rieties, is iterated to a comprehensive class of projective limits of
such varieties. This framework includes as necessary ingredients
recent works on the multivariate moment problem, disintegration
and projective limits of probability measures and basic techniques
of the theory of locally convex vector spaces. A variety of applica
tions illustrate the versatility of this novel geometric approach to
polynomial optimization.
1. Introduction
The ubiquous duality between ideals and algebraic varieties is re
placed in semialgebraic geometry by a duality between preorders, or
quadratic modules in a ring (see the preliminaries for the exact defini
tions) and their positivity sets. This is already a nontrivial departure
from classical algebraic geometry, well studied and understood only
in the last decades with tools from real algebra, logic and functional
