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

TL;DR

This paper investigates the Hermitian null-range of matrices over finite fields, focusing on the sets of values generated by the Hermitian form for vectors with specific properties, especially in the case of two-dimensional matrices.

Contribution

It provides a detailed analysis of the Hermitian null-range for matrices over finite fields, including explicit descriptions for 2x2 matrices and cases where the field characteristic is even.

Findings

Explicit description of Num_0(M) and Num_0(M)_q for 2x2 matrices.

Simplified characterization of Num_0(M)_q when the field characteristic is even.

Extension of previous work on numerical ranges to finite field settings.

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 $n$.