Real Algebraic and Analytic Geometry Preprint Server

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Submission: 2004, August 28.

Abstract:
Let g_1,...,g_r be real polynomials in n variables such that the set K = {g_1>=0,...,g_r>=0} in IR^n is compact. We study the problem of representing polynomials f with f>=0 on K in the form f = s_0+s_1g_1+ ...+s_rg_r with sums of squares s_i, with particular emphasis on the case where f has zeros in K. Assuming that the quadratic module of all such sums is archimedean, we establish a local-global condition for f to have such a representation, vis-a-vis the zero set of f in K. This criterion is most useful when f has only finitely many zeros in K. We present a number of concrete situations where this result can be applied. As another application we solve an open problem from [KMS] on one-dimensional quadratic modules. .

Mathematics Subject Classification (2000): 14P05, 11E25, 14P10, 26D05.

Keywords and Phrases: Non-negative polynomials, sums of squares, positivity, quadratic modules, semiorderings, real algebraic geometry.

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