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

TL;DR

This paper proves that every sum of squares in the three-variable Laurent series field over the reals is a sum of four squares, extending previous results and generalizing to arbitrary real fields, with implications for the $u$-invariant.

Contribution

It establishes the minimal number of squares needed to represent sums of squares in three-variable Laurent series fields and generalizes known results to arbitrary real fields.

Findings

Every sum of squares in $ ext{R}((x,y,z))$ is a sum of 4 squares.

Every sum of squares in finite extensions of $ ext{R}((x,y))$ is a sum of 3 squares.

Generalization of sum of squares results to arbitrary real fields.

Abstract

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.