On Matrix Algebras Isomorphic to Finite Fields and Planar Dembowski-Ostrom Monomials

Let $p$ be a prime and $n$ a positive integer. As the first main result, we present a deterministic algorithm for deciding whether the matrix algebra $\mathbb{F}_p[A_1,\dots,A_t]$ with $A_1,\dots,A_t \in \mathrm{GL}(n,\mathbb{F}_p)$ is a finite field, performing at most $\mathcal{O}(tn^6\log(p))$ elementary operations in $\mathbb{F}_p$. In the affirmative case, the algorithm returns a defining element $a$ so that $\mathbb{F}_p[A_1,\dots,A_t] = \mathbb{F}_p[a]$. We then study an invariant for the extended-affine equivalence of Dembowski-Ostrom (DO) polynomials. More precisely, for a DO polynomial $g \in \mathbb{F}_{p^n}[x]$, we associate to $g$ a set of $n \times n$ matrices with coefficients in $\mathbb{F}_p$, denoted $\mathrm{Quot}(\mathcal{D}_g)$, that stays invariant up to matrix similarity when applying extended-affine equivalence transformations to $g$. In the case where $g$ is a planar DO polynomial, $\mathrm{Quot}(\mathcal{D}_g)$ is the set of quotients $XY^{-1}$ with $Y \neq 0,X$ being elements from the spread set of the corresponding commutative presemifield, and $\mathrm{Quot}(\mathcal{D}_g)$ forms a field of order $p^n$ if and only if $g$ is equivalent to the planar monomial $x^2$, i.e., if and only if the commutative presemifield associated to $g$ is isotopic to a finite field. As the second main result, we analyze the structure of $\mathrm{Quot}(\mathcal{D}_g)$ for all planar DO monomials, i.e., for commutative presemifields of odd order being isotopic to a finite field or a commutative twisted field. More precisely, for $g$ being equivalent to a planar DO monomial, we show that every non-zero element $X \in \mathrm{Quot}(\mathcal{D}_g)$ generates a field $\mathbb{F}_p[X] \subseteq \mathrm{Quot}(\mathcal{D}_g)$ and $\mathrm{Quot}(\mathcal{D}_g)$ contains the field $\mathbb{F}_{p^n}$.