Real Algebraic and Analytic Geometry
Submission: 2005, September 1.
Among the invariant factors $g$ of a positive semidefinite analytic function $f$ on $\R^3$, those $g$ whose zero set $Y$ is a curve are called \em special\em. We show that if each special $g$ is a sum of squares of global meromorphic functions on a neighbourhood of $Y$, then $f$ is a sum of squares of global meromorphic functions. Here sums can be (convergent) infinite, and we also find some sufficient conditions to get \em finite \em sums of squares. In addition, we construct several examples of positive semidefinite analytic functions which are infinite sums of squares but maybe could not be finite sum of squares.
Mathematics Subject Classification (2000): 14P99, 11E25, 32B10, 32S05.
Keywords and Phrases: 17th Hilbert Problem, sum of squares, irreducible factors, special factors.
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