Sums of squares in function fields over henselian discretely valued fields
Number Theory
2023-07-03 v2
Abstract
Let and let be a field with a henselian discrete valuation of rank with hereditarily euclidean residue field. Let be an algebraic function field in one variable. We show that the Pythagoras number of is or and we determine the order of the group of nonzero sums of squares modulo sums of two squares in in terms of the number of equivalence classes of discrete valuations on of rank at most In the case of function fields of hyperelliptic curves of genus , K.J. Becher and J. Van Geel showed that the order of this quotient group is bounded by . We show in Example 4.6 that this bound is optimal.
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}
}