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 $\mathbb {F} _{q^2}$. In this paper, it is considered the case $k=0$ of the set $\mathrm{Num} _k(M):= \{\langle u,Mu\rangle \mid u\in \mathbb {F} _{q^2}^n, \langle u,u\rangle =k\}$. When $M$ has coefficients in $\mathbb {F} _q$ the paper studies the set $\mathrm{Num} _k(M)_q:= \{\langle u,Mu\rangle \mid u\in \mathbb {F} _q^n,\langle u,u\rangle =k\}\subseteq \mathbb {F} _q$. 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 the author (arbitrary $q$). In this paper, it is studied in details $\mathrm{Num} _0(M)$ and $\mathrm{Num} _k(M)_q$ when $n=2$. If $q$ is even, $\mathrm{Num} _0(M)_q$ is easily described for arbitrary $n$. If $q$ is odd, then either $\mathrm{Num} _0(M)_q =\{0\}$, or $\mathrm{Num} _0(M)_q=\mathbb {F} _q$, or $\sharp (\mathrm{Num} _0(M)_q)=(q+1)/2$.