Nonsimilar half-neighbors over fields of characteristic 2

Abstract

The total isotropy index of a quadratic form $\varphi$ over a field $F$ is the maximum dimension of any totally isotropic subspace of $\varphi$. If $\varphi$ is anisotropic and $\psi$ is another anisotropic quadratic form over $F$ of the same dimension, then $\varphi$ and $\psi$ are called Vishik-equivalent if, over any field extension $E/F$, their total isotropy indices are the same. In characteristic $\neq 2$, Vishik-equivalence implies similarity in all dimensions $\leq 7$ and in all odd dimensions, but there are counterexamples in all even dimensions $\geq 8$. In this paper, we construct semi-singular anisotropic quadratic forms of dimension $2^m$ for any $m\geq 3$ and defined over a suitable extension of any given field $F_0$ of characteristic $2$ that are Vishik-equivalent but not similar, thus completing the list of such examples provided earlier by the first author and Krist\'yna Zemkov\'a.