The Hermitian null-range of a matrix over a finite field

Abstract: Let $q$ be a prime power. For $u=(u_1,\dots ,u_n), v=(v_1,\dots ,v_n)\in \mathbb {F}_{q^2}^n$ let $\langle u,v\rangle := \sum _{i=1}^{n} u_i^qv_i$ be the Hermitian form of $\mathbb {F} _{q^2}^n$. Fix an $n\times n$ matrix $M$ over $\mathbb {F} _{q^2}$. We study the case $k=0$ of the set $\mathrm{Num} _k(M):= {\langle u,Mu\rangle \mid u\in \mathbb {F} _{q^2}, \langle u,u\rangle =k}$. When $M$ has coefficients in $\mathbb {F} _q$ we study the set $\mathrm{Num} _0(M)_q:= {\langle u,Mu\rangle \mid u\in \mathbb {F} _q^n}\subseteq \mathbb {F} _q$. The set $\mathrm{Num} _1(M)$ is the numerical range of $M$, previously introduced in a paper by Coons, Jenkins, Knowles, Luke and Rault (case $q$ a prime $p\equiv 3\pmod{4}$) and by myself (arbitrary $q$). We study in details $\mathrm{Num} _0(M)$ and $\mathrm{Num} _0(M)_q$ when $n=2$. If $q$ is even, $\mathrm{Num} _0(M)_q$ is easily described for arbitrary $n$.