In this talk, we consider symmetric functions which, under the truncation of variables, can be identified with sequences of symmetric polynomials in all numbers of variables. We call a symmetric function nonnegative (positive) if all associated symmetric polynomials are nonnegative (positive). In general, deciding whether symmetric functions are nonnegative is non-trivial and actually requires verification of a polynomial's nonnegativity on a set that is not semialgebraic. In recent work by Chandrasekaran and Levin such problems have been called any-dimensional problems and it was asked whether Positivstellensätze for this setting exist. As a first step we prove certain analogues of Pólya's and Reznick's Positivstellensätze.

We investigate foundational questions in the study of tropicalizations of semialgebraic sets, and more generally of definable sets in a polynomially bounded o-minimal structure. Our motivations come from the study of the complexity of representations of positive polynomials, from the sudy of bounded ratios for Lorenzian matrices, and from Strassen’s theory of the asymptotic spectrum.

We use techniques from model theory and real algebraic geometry to prove the Foundamental Theorem of Tropical O-minimal Geometry. We discuss initial degenerations and initial preorders, and apply our results to study the geometry of (irrational) toric varieties.

Based on a joint work with M. Telek.