The Pythagoras number and the u-invariant of Laurent series fields in several variables

We show that every sum of squares in the three-variable Laurent series field R((x,y,z))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 R((x,y))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 R((x,y))R((x,y)) itself is a sum of two squares. We give a generalization of this result where RR is replaced by an arbitrary real field. Our methods yield similar results about the u-invariant of fields of the same type.