Real Algebraic and Analytic Geometry

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185. Francesca Acquistapace, Fabrizio Broglia, José F. Fernando, Jesús M. Ruiz:
On the finiteness of Pythagoras numbers of real meromorphic functions.

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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.

Full text, 16p.: pdf 162k.

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