The Pythagoras number of a rational function field in two variables

TL;DR

This paper establishes that in the rational function field in two variables over a hereditarily pythagorean field, every sum of squares can be expressed as a sum of at most eight squares, with implications for quadratic form theory.

Contribution

It proves a new upper bound on the Pythagoras number of rational function fields over hereditarily pythagorean fields, extending understanding of sums of squares in such fields.

Findings

Sum of squares in $K(X,Y)$ is a sum of 8 squares.

Pythagoras number of finite extensions of $K(X)$ is at most 5.

Utilizes local-global principles and valuation theory for proofs.

Abstract

We prove that every sum of squares in the rational function field in two variables $K(X,Y)$ over a hereditarily pythagorean field $K$ is a sum of $8$ squares. More precisely, we show that the Pythagoras number of every finite extension of $K(X)$ is at most $5$. The main ingredients of the proof are a local-global principle for quadratic forms over function fields in one variable over a complete rank-$1$ valued field due to V. Mehmeti and a valuation theoretic characterization of hereditarily pythagorean fields due to L. Br\"ocker.