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

### José F. Fernando

Universidad Complutense de Madrid, Spain

## 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.

## Cite this article

José F. Fernando, On Hilbert’s 17th Problem for global analytic functions in dimension 3. Comment. Math. Helv. 83 (2008), no. 1, pp. 67–100

DOI 10.4171/CMH/119