Quasilinear quadratic forms and function fields of quadrics

Abstract

Let $p$ and $q$ be anisotropic quadratic forms of dimension $\geq 2$ over a field $F$. In a recent article, we formulated a conjecture describing the general constraints which the dimensions of $p$ and $q$ impose on the isotropy index of $q$ after scalar extension to the function field of $p$. This can be viewed as a generalization of Hoffmann's Separation Theorem which simultaneously incorporates and refines some well-known classical results on the Witt kernels of function fields of quadrics. Using algebro-geometric methods, it was shown that large parts of this conjecture hold in the case where the characteristic of $F$ is not 2. In the present article, we prove similar (in fact, slightly stronger) results in the case where $F$ has characteristic $2$ and $q$ is a so-called quasilinear form. In contrast to the situation where $\mathrm{char}(F) \neq 2$, the methods used to treat this case are purely algebraic.