Quasilinear quadratic forms and function fields of quadrics

Abstract

Let p and q be anisotropic quadratic forms of dimension $$\ge 2$$ over a field F. In a recent article, we formulated a conjecture describing a general constraint 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, somewhat sharper) results in the case where F has characteristic 2 and q is a so-called quasilinear form. In contrast to the situation where $$\mathrm {char}(F) \ne 2$$, the methods used to treat this case are purely algebraic.