The Pythagoras number of fields of transcendence degree $1$ over $\mathbb{Q}$

TL;DR

This paper proves that in fields of transcendence degree one over the rationals, any sum of squares can be expressed as a sum of five squares, resolving a question posed by Pop and Pfister.

Contribution

It establishes a new bound on the Pythagoras number for these fields by proving all sums of squares are sums of five squares, based on a representation theorem for quadratic forms.

Findings

Any sum of squares in such fields is a sum of 5 squares

Derived from a quadratic form representation theorem over curves

Answers a question of Pop and Pfister

Abstract

We show that any sum of squares in a field of transcendence degree $1$ over $\mathbb{Q}$ is a sum of $5$ squares, answering a question of Pop and Pfister. We deduce this result from a representation theorem, in $k(C)$, for quadratic forms of rank $\geq 5$ with coefficients in $k$, where $C$ is a curve over a number field $k$.