On the splitting of quasilinear $p$-forms

TL;DR

This paper investigates the splitting behavior of quasilinear p-forms in characteristic p fields, extending classical quadratic form theory with new applications and analogues of key theorems in motives and invariants.

Contribution

It introduces an analogue of Knebusch's generic splitting tower for quasilinear p-forms and provides new results including an algebraic analogue of Vishik's theorem and partial progress on Karpenko's invariant.

Findings

Established an algebraic analogue of Vishik's theorem on motives of quadrics.

Obtained partial results towards a quasilinear analogue of Karpenko's theorem.

Proved a conjecture of Hoffmann on maximal splitting of quasilinear quadratic forms.

Abstract

We study the splitting behaviour of quasilinear $p$-forms in the spirit of the theory of nondegenerate quadratic forms over fields of characteristic different from 2 using an analogue of M. Knebusch's generic splitting tower. Several new applications to the theory of quasilinear quadratic forms are given. Among them, we can mention an algebraic analogue of A. Vishik's theorem on "outer excellent connections" in the motives of quadrics, partial results towards a quasilinear analogue of N. Karpenko's theorem on the possible values of the invariant $i_1$, and a proof of a conjecture of D. Hoffmann on quadratic forms with maximal splitting in the quasilinear case.