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- W2559443382 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 $langle u,vrangle := sum _{i=1}^{n} u_i^qv_i$ be the Hermitian form of $mathbb {F} _{q^2}^n$. Fix an $ntimes n$ matrix $M$ over $mathbb {F} _{q^2}$. In this paper, it is considered the case $k=0$ of the set $mathrm{Num} _k(M):= {langle u,Murangle mid uin mathbb {F} _{q^2}^n, langle u,urangle =k}$. When $M$ has coefficients in $mathbb {F} _q$ the paper studies the set $mathrm{Num} _k(M)_q:= {langle u,Murangle mid uin mathbb {F} _q^n,langle u,urangle =k}subseteq mathbb {F} _q$. 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 $pequiv 3pmod{4}$), 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 $sharp (mathrm{Num} _0(M)_q)=(q+1)/2$." @default.
- W2559443382 created "2016-12-08" @default.
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- W2559443382 date "2018-02-21" @default.
- W2559443382 modified "2023-09-24" @default.
- W2559443382 title "The Hermitian Null-range of a Matrix over a Finite Field" @default.
- W2559443382 doi "https://doi.org/10.13001/1081-3810.3416" @default.
- W2559443382 hasPublicationYear "2018" @default.
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