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arXiv · 2203.14357

Sums of squares in function fields over henselian discretely valued fields

Gonzalo Manzano-Flores
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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.

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Gonzalo Manzano-Flores. 2022-03-27. Sums of squares in function fields over henselian discretely valued fields. https://arxiv.org/abs/2203.14357

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