The u-invariant of function fields in one variable
Abstract
The u-invariant of a field is the largest dimension of an anisotropic quadratic torsion form over the field. In this article we obtain a bound on the u-invariant of function fields in one variable over a henselian valued field with arbitrary value group and with residue field of characteristic different from 2. This generalises a theorem due to Harbater, Hartmann and Krashen and its extension due to Scheiderer. Their result covers the special case where the valuation is discrete. We further give a new proof of a theorem due to Parimala and Suresh bounding by 8 the u-invariant of a function field in one variable over any henselian discretely valued field of characteristic 0 with perfect residue field of characteristic 2.
Keywords
Cite
@article{arxiv.2502.13086,
title = {The u-invariant of function fields in one variable},
author = {Karim Johannes Becher and Nicolas Daans and Vlerë Mehmeti},
journal= {arXiv preprint arXiv:2502.13086},
year = {2025}
}
Comments
33 pages; v2 includes an additional section containing first steps towards the study of fields with residue characteristic 2, several changes were made throughout the paper to incorporate this case