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Numerical ranges over a certain family of finite fields were classified in 2016 by a team including our fifth author. Soon afterward, in 2017 Ballico generalized these results to all finite fields and published some new results about the cardinality of the finite field numerical range. In this paper we study the geometry of these finite fields using the boundary generating curve, first introduced by Kippenhahn in 1951. We restrict our study to square matrices of dimension 2, with at least one eigenvalue in $mathbb F_{q^2}$.
In the paper [On superspecial abelian surfaces over finite fields II. J. Math. Soc. Japan, 72(1):303--331, 2020], Tse-Chung Yang and the first two current authors computed explicitly the number $lvert mathrm{SSp}_2(mathbb{F}_q)rvert$ of isomorphism c
If the $ell$-adic cohomology of a projective smooth variety, defined over a $frak{p}$-adic field $K$ with finite residue field $k$, is supported in codimension $ge 1$, then any model over the ring of integers of $K$ has a $k$-rational point. This sli
In this paper we introduce the additive analogue of the index of a polynomial over finite fields. We study several problems in the theory of polynomials over finite fields in terms of their additive indices, such as value set sizes, bounds on multipl
Let ${mathbb F}_q$ be the finite field with $q=p^k$ elements with $p$ being a prime and $k$ be a positive integer. For any $y, zinmathbb{F}_q$, let $N_s(z)$ and $T_s(y)$ denote the numbers of zeros of $x_1^{3}+cdots+x_s^3=z$ and $x_1^3+cdots+x_{s-1}^
Let $mathbb{F}_q$ be the finite field of $q=p^mequiv 1pmod 4$ elements with $p$ being an odd prime and $m$ being a positive integer. For $c, y inmathbb{F}_q$ with $yinmathbb{F}_q^*$ non-quartic, let $N_n(c)$ and $M_n(y)$ be the numbers of zeros of $x