Quasilinear quadratic forms and function fields of quadrics

TL;DR

This paper extends the understanding of quadratic forms and their isotropy over function fields, particularly in characteristic 2, by proving new algebraic results for quasilinear forms that generalize classical theorems.

Contribution

It proves new algebraic results on the isotropy of quasilinear quadratic forms over function fields in characteristic 2, strengthening previous geometric results for non-2 characteristic fields.

Findings

Proved stronger results for quasilinear forms in characteristic 2.

Extended classical theorems to the case of characteristic 2.

Demonstrated algebraic methods can replace geometric ones in this context.

Abstract

Let $p$ and $q$ be anisotropic quadratic forms of dimension $\geq 2$ over a field $F$. In a recent article, we formulated a conjecture describing the general constraints which the dimensions of $p$ and $q$ impose on the isotropy index of $q$ after scalar extension to the function field of $p$. This can be viewed as a generalization of Hoffmann's Separation Theorem which simultaneously incorporates and refines some well-known classical results on the Witt kernels of function fields of quadrics. Using algebro-geometric methods, it was shown that large parts of this conjecture hold in the case where the characteristic of $F$ is not 2. In the present article, we prove similar (in fact, slightly stronger) results in the case where $F$ has characteristic $2$ and $q$ is a so-called quasilinear form. In contrast to the situation where $\mathrm{char}(F) \neq 2$, the methods used to treat this case are purely algebraic.