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

Kristýna Zemková

Vol. 329 (2024), No. 2, 327–356

DOI: 10.2140/pjm.2024.329.327

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 quasilinear $p$-forms. We study the question whether Vishik equivalent $p$-forms must be similar. 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 a positive answer.

Keywords

quasilinear $p$-forms, quadratic forms, finite characteristic, isotropy, equivalence relations

Mathematical Subject Classification

Primary: 11E04, 11E81

Milestones

Received: 12 February 2024

Revised: 30 May 2024

Accepted: 14 June 2024

Published: 6 July 2024