Vishik equivalence and similarity of quasilinear $p$-forms and totally singular quadratic forms

TL;DR

This paper extends the concept of Vishik equivalence from quadratic forms over fields of characteristic not two to quasilinear p-forms, exploring conditions under which equivalence implies similarity, especially for totally singular quadratic forms.

Contribution

It introduces Vishik equivalence for quasilinear p-forms and investigates when this equivalence implies similarity, providing new insights for totally singular quadratic forms.

Findings

Vishik equivalence does not always imply similarity for quasilinear p-forms.

For certain families of totally singular quadratic forms, Vishik equivalence does imply similarity.

The paper establishes conditions under which Vishik equivalence and similarity coincide for these forms.

Abstract

For quadratic forms over fields of characteristic different from two, there is a so-called Vishik criterion, giving a purely algebraic characterization of when two quadratic forms are motivically equivalent. In analogy to that, we define Vishik equivalence on quasiliner $p$-forms. We study the question whether Vishik equivalent $p$-forms must be similiar. We prove that this is not true for quasilinear $p$-forms in general, but we find some families of totally singular quadratic forms (i.e., of quasilinear $2$-forms) for which the question has positive answer.