On Hilbert’s 17th Problem for global analytic functions in dimension 3

Abstract

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 special. 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, but we also find some sufficient conditions to get finite 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 sums of squares.