Real Algebraic and Analytic Geometry
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Submission: 2005, December 9.
We prove that if Hilbert's 17th Problem for global analytic functions has a positive solution for the affine space, then the Pythagoras numbers of the rings (i) of global meromorphic functions, and (ii) of meromorphic function germs, are both finite. This is a measure of the difficulty of the problem for analytic functions in the non-compact case.
Mathematics Subject Classification (2000): 14P99, 11E25, 32B10, 32S05.
Keywords and Phrases: Hilbert's 17th Problem, Pythagoras number, infinite sum of squares, bad set, germs at closed sets.
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