On the splitting of quasilinear $p$-forms

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.