  We show that every sum of squares in the three-variable Laurent series field
$\mathbb{R}((x,y,z))$ is a sum of 4 squares, as was conjectured in a paper of
Choi, Dai, Lam and Reznick in the 1980's. We obtain this result by proving that
every sum of squares in a finite extension of $\mathbb{R}((x,y))$ is a sum of
$3$ squares. It was already shown in Choi, Dai, Lam and Reznick's paper that
every sum of squares in $\mathbb{R}((x,y))$ itself is a sum of two squares. We
give a generalization of this result where $\mathbb{R}$ is replaced by an
arbitrary real field. Our methods yield similar results about the $u$-invariant
of fields of the same type.
