Real Algebraic and Analytic Geometry Preprint Server

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Submission: 2004, August 28.

Abstract:
Consider real polynomials g_1,...,g_r in n variables, and assume that the subset K = {g_1>=0,...,g_r>=0} of R^n is compact. We show that a polynomial f has a representation f = \sum_{e\in\{0,1\}^r} s_e.g_1^{e_1}...g_r^{e_r} (*) in which the s_e are sums of squares, if and only if the same is true in every localization of the polynomial ring by a maximal ideal. We apply this result to provide large and concrete families of cases in which dim(K)=2 and every polynomial f with f|K >= 0 has a representation (*). Before, it was not known whether a single such example exists. Further geometric and arithmetic applications are given .

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

Keywords and Phrases: Non-negative polynomials, polynomial inequalities, sums of squares, preorderings, archimedean, real algebraic geometry, real algebraic surfaces, Hilbert 17th problem.

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