Sums of squares in function fields over henselian discretely valued fields

Abstract

Let $n\in N$ and let K be a field with a henselian discrete valuation of rank n with hereditarily euclidean residue field. Let $F/K$ be a function field in one variable. It is known that every sum of squares is a sum of 3 squares. We determine the order of the group of nonzero sums of 3 squares modulo sums of 2 squares in F in terms of equivalence classes of certain discrete valuations of 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 that this bound is optimal. Moreover, in the case where $n=1$, we show that if $F/K$ is a hyperelliptic function field such that the order of this quotient group is $2^{g+1}$, then F is nonreal.