On the finiteness of Pythagoras numbers of real meromorphic functions.

Abstract

We consider the 17(th) Hilbert Problem for global real analytic functions in a modified form that involves infinite sums of squares. Then we prove a local-global principle for a real global analytic function to be a sum of squares of global real meromorphic functions. We deduce that an affirmative solution to the 17(th) Hilbert Problem for global real analytic functions implies the finiteness of the Pythagoras number of the field of global real meromorphic functions, hence that of the field of real meromorphic power series. This measures the difficulty of the 17(th) Hilbert problem in the analytic case.