Positive Polynomials and Sequential Closures of Quadratic Modules

Let S = {x ∈ Rn | f1(x) ≥ 0, . . . , fs(x) ≥ 0} be a basic closed semi-algebraic set in Rn and let PO(f1, . . . , fs) be the corresponding preordering in R[X1, . . . , Xn]. We examine for which polynomials f there exist identities f + εq ∈ PO(f1, . . . , fs) for all ε > 0. These are precisely the elements of the sequential closure of PO(f1, . . . , fs) 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 PO(f1, . . . , fs) ∨∨, 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 fibrepreorderings, 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 PO(f1, . . . , fs)∨∨ do. The results also hold in the more general setup of arbitrary commutative algebras and quadratic modules instead of preorderings.