Bounded Polynomials, Sums of Squares, and the Moment Problem

Abstract

We study representations of positive polynomial functions as sums of squares or weighted sums of squares (in so-called preorderings). Approximating positive polynomials by such representations is related to the moment problem from functional analysis. In particular, we study the geometry of bounded polynomial functions in connection with a recent theorem of Schmüdgen that gives a necessary and sufficient condition for the moment problem on a semialgebraic set in terms of the moment problem on the fibres of a bounded polynomial map.In Chapter 1, we show how the ring of bounded polynomial functions on a semialgebraic set can (under suitable conditions) be represented as the ring of regular functions on a quasiprojective variety. As an application, we use a theorem of Zariski to show that the ring of bounded functions on a sufficiently regular two-dimesional set is a finitely generated R-algebra.Chapter 2 contains references to known results about positivity and sums of squares. In particular, we prove a stronger version of the Powers-Scheiderer theorem concerning the existence of degree bounds for preorderings.Chapter 3 is devoted to the case of curves. Here, we generalise results of Scheiderer from the irreducible to the reducible case. We classify all real curves having the moment property. We also prove a result concerning the extension of psd functions from curves to ambient space.In Chapter 4, we apply the techniques from the previous chapters to the moment problem and sums of squares in the two-dimensional case. We also reprove a theorem of Roggero concerning the divisor class group of a real variety.