Positive polynomials and sequential closures of quadratic modules

Abstract:

Let $\mathcal {S}=\{x\in \mathbb {R}^n\mid f_1(x)\geq 0,\ldots ,f_s(x)\geq 0\}$ be a basic closed semi-algebraic set in $\mathbb {R}^n$ and let $\mathrm {PO}(f_1,\ldots ,f_s)$ be the corresponding preordering in $\mathbb {R}[X_1,\ldots ,X_n]$. We examine for which polynomials $f$ there exist identities \[ f+\varepsilon q\in \mathrm {PO}(f_1,\ldots ,f_s) \mbox { for all } \varepsilon >0.\] These are precisely the elements of the sequential closure of $\mathrm {PO}(f_1,\ldots ,f_s)$ with respect to the finest locally convex topology. We solve the open problem from Kuhlmann, Marshall, and Schwartz (2002, 2005), whether this equals the double dual cone \[ \mathrm {PO}(f_1,\ldots ,f_s)^{\vee \vee },\] by providing a counterexample. We then prove a theorem that allows us 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 $\mathcal {S}$ admits such representations, or at least the polynomials from $\mathrm {PO}(f_1,\ldots ,f_s)^{\vee \vee }$ do. The results also hold in the more general setup of arbitrary commutative algebras and quadratic modules instead of preorderings.