English

Sums of squares in function fields over henselian discretely valued fields

Number Theory 2023-07-03 v2

Abstract

Let nNn\in\mathbb{N} and let KK be a field with a henselian discrete valuation of rank nn with hereditarily euclidean residue field. Let F/KF/K be an algebraic function field in one variable. We show that the Pythagoras number of FF is 22 or 33 and we determine the order of the group of nonzero sums of squares modulo sums of two squares in FF in terms of the number of equivalence classes of discrete valuations on FF of rank at most n.n. In the case of function fields of hyperelliptic curves of genus gg, K.J. Becher and J. Van Geel showed that the order of this quotient group is bounded by 2n(g+1)2^{n(g+1)}. We show in Example 4.6 that this bound is optimal.

Keywords

Cite

@article{arxiv.2203.14357,
  title  = {Sums of squares in function fields over henselian discretely valued fields},
  author = {Gonzalo Manzano-Flores},
  journal= {arXiv preprint arXiv:2203.14357},
  year   = {2023}
}
R2 v1 2026-06-24T10:27:32.605Z