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Positive polynomials and sequential closures of quadratic modules

By Tim Netzer


Abstract. 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. 1

Year: 2013
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