The u-invariant of function fields in one variable

TL;DR

This paper establishes bounds on the u-invariant for function fields in one variable over henselian valued fields, extending previous results to more general valuation groups and residue fields, and provides new proofs for specific cases.

Contribution

It generalizes existing theorems on the u-invariant by considering arbitrary value groups and residue fields, and offers a new proof for bounds over discretely valued fields of characteristic 0.

Findings

Bound on u-invariant over henselian valued fields with arbitrary value group

Extension of Harbater, Hartmann, Krashen, and Scheiderer's results

New proof bounding u-invariant by 8 over discretely valued fields of characteristic 0

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.