Sums of squares in function fields over a henselian valued field

Abstract

The Pythagoras number of a field is the smallest natural number $n$ such that every sum of squares in the field is a sum of $n$ squares. Given a nondyadic henselian valued field $(K,v)$, we show that the best bound on the Pythagoras number for function fields of curves over $K$ is essentially the same as for function fields of curves over the residue field $Kv$. As a consequence, we obtain the upper bound $5$ for the Pythagoras number of function fields of curves over any field $K$ for which the rational function field $K(X)$ has Pythagoras number $2$. The proof uses a reduction to the case where the value group is a subgroup of the real numbers, where then a local-global principle for quadratic forms due to V.~Mehmeti is applied.

27 pages

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