Universal quadratic forms over semi-global fields

TL;DR

This paper investigates the properties of anisotropic universal quadratic forms over semi-global fields, computing invariants and establishing a local-global principle to understand their behavior.

Contribution

It introduces the computation of the $m$-invariant and the set of dimensions of anisotropic universal quadratic forms over semi-global fields, and defines the strong $m$-invariant.

Findings

Computed the $m$-invariant for semi-global fields

Determined the possible dimensions of anisotropic universal quadratic forms

Established a local-global principle for isotropy of quadratic forms

Abstract

We study anisotropic universal quadratic forms over semi-global fields; i.e., over one-variable function fields over complete discretely valued fields. In particular, given a semi-global field $F$, we compute both the $m$-invariant of $F$ and the set of dimensions of anisotropic universal quadratic forms over $F$. We also define the strong $m$-invariant of a field $k$ and show that it behaves analogously to the strong $u$-invariant of $k$, defined by Harbater, Hartmann, and Krashen. Our main tool in this study is the local-global principle for isotropy of quadratic forms over a semi-global field with respect to particular sets of overfields.