Positive Polynomials and Sequential Closures of Quadratic Modules

TL;DR

This paper investigates the conditions under which polynomials can be approximated by elements in quadratic modules, providing counterexamples and new methods to represent nonnegative polynomials in semi-algebraic sets.

Contribution

It introduces a novel approach using fibre-preorderings to analyze polynomial representations and resolves an open problem regarding the equality of the sequential closure and the double dual cone.

Findings

Counterexample showing the sequential closure does not always equal the double dual cone

A new theorem linking fibre-preorderings to polynomial identities

Conditions under which nonnegative polynomials admit specific representations

Abstract

Let S be a basic closed semi-algebraic set in R^n and P the corresponding preordering in R[X_1,...,X_n]. We examine for which polynomials f there exist identities f+\ep q \in P for all \ep>0. These are precisely the elements of the sequential closure of P with respect to the finest locally convex topology. We solve the open problem whether this equals the double dual cone of P, by providing a counterexample. We then prove a theorem that allows to obtain identities for polynomials as above, by looking at a family of fibre-preorderings, constructed from bounded polynomials. These fibre-preorderings are easier to deal with than the original preordering in general. For a large class of examples we are thus able to show that either every polynomial f that is nonnegative on S admits such representations, or at least the polynomials from the double dual cone of P do. The results also hold in the more general setup of arbitrary commutative algebras and quadratic modules instead of preorderings.