Extended Karpenko and Karpenko-Merkurjev theorems for quasilinear quadratic forms

Let p and q be anisotropic quasilinear quadratic forms over a field F of characteristic $2$, and let i be the isotropy index of q after scalar extension to the function field of the affine quadric with equation $p=0$. In this article, we establish a strong constraint on i in terms of the dimension of q and two stable birational invariants of p, one of which is the well-known "Izhboldin dimension", and the other of which is a new invariant that we denote Δ(p). Examining the contribution from the Izhboldin dimension, we obtain a result that unifies and extends the quasilinear analogues of two fundamental results on the isotropy of non-singular quadratic forms over function fields of quadrics in arbitrary characteristic due to Karpenko and Karpenko-Merkurjev, respectively. This proves in a strong way the quasilinear case of a general conjecture previously formulated by the author, suggesting that a substantial refinement of this conjecture should hold.