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 $⟨u,v⟩:={\sum }_{i=1}^n{u}_i^q{v}_i$ 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 ${\mathrm{Num}}_k(M):=\{⟨u,Mu⟩\mid u\in {\mathbb{F}}_{q^2},⟨u,u⟩=k\}$. When $M$ has coefficients in ${\mathbb{F}}_q$ we study the set ${\mathrm{Num}}_0(M{)}_q:=\{⟨u,Mu⟩\mid u\in {\mathbb{F}}_q^n\}\subseteq {\mathbb{F}}_q$. 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 $p\equiv 3(\mathrm{mod}4)$) 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 $n$.