Polynomials non-negative on strips and half-strips

An expression f = σ+ τ(x− x) is an immediate witness to the positivity condition on f . In general, one wants to characterize polynomials f which are positive, or nonnegative, on a semialgebraic set K ⊆ R in terms of sums of squares and the polynomials used to define K. Representation theorems of this type have a long and illustrious history, going back at least to Hilbert. There has been much interest in these questions in the last decade, in a large part because of applications outside of real algebraic geometry, notably in problems of optimizing polynomial functions on semialgebraic sets. In this paper we look at some generalizations of Marshall’s theorem. Our results give many new examples of non-compact semialgebraic sets in R for which one can characterize all polynomials which are non-negative on the set. Let R[X] denote R[x1, . . . , xn], the real polynomial ring in n variables, and write ∑ R[X] for the sums of squares in R[X]. Given a finite set S = {s1, . . . , sk} ⊆ R[X] the basic closed semialgebraic set in R generated by S, denoted KS , is {a ∈ R | si(a) ≥ 0 for i = 1, . . . , k}. Note that the strip [0, 1] × R is the basic closed semialgebraic set in R generated by {x− x}. There are two algebraic objects associated to the semialgebraic set KS : The quadratic module generated by S, denoted MS , is the set of all elements of R[X] which can be written σ0 + σ1s1 + · · · + σksk, where each σi ∈ ∑ R[X]. The preordering generated by S, denoted TS , consists of all elements of the form ∑ e∈{0,1}k σes , where s denotes s1 1 . . . s es s for e = (e1, . . . , es), and each σe ∈ ∑ R[X]. In general, MS $ TS , although if |S| = 1, then clearly TS = MS . Also, TS = MS iff MS is closed under multiplication iff si · sj ∈MS for all i, j. ∗Department of Mathematics, Wesleyan College, Macon, GA 31210. Email: [email protected] . †Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322. Email: [email protected].