Real Algebraic and Analytic Geometry |

On the finiteness of Pythagoras numbers of real meromorphic functions.

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Submission: 2005, December 9.

*Abstract:
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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