Sums of squares in function fields over henselian discretely valued fields

TL;DR

This paper investigates sums of squares in function fields over henselian discretely valued fields, determining the Pythagoras number and the structure of sums of squares modulo sums of two squares, with explicit bounds and examples.

Contribution

It establishes the Pythagoras number as 2 or 3 for these fields and determines the order of the sum of squares group, extending previous bounds and providing examples of optimality.

Findings

Pythagoras number of the function field is 2 or 3.

The order of the sum of squares quotient group is characterized by valuations.

Bound for hyperelliptic curves is shown to be optimal.

Abstract

Let $n\in\mathbb{N}$ and let $K$ be a field with a henselian discrete valuation of rank $n$ with hereditarily euclidean residue field. Let $F/K$ be an algebraic function field in one variable. We show that the Pythagoras number of $F$ is $2$ or $3$ and we determine the order of the group of nonzero sums of squares modulo sums of two squares in $F$ in terms of the number of equivalence classes of discrete valuations on $F$ of rank at most $n.$ In the case of function fields of hyperelliptic curves of genus $g$, K.J. Becher and J. Van Geel showed that the order of this quotient group is bounded by $2^{n(g+1)}$. We show in Example 4.6 that this bound is optimal.