On matrix algebras isomorphic to finite fields and planar Dembowski-Ostrom monomials

Abstract

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 $F_p[A_1,\dots ,A_t]$ with $A_1,\dots ,A_t\in \mathrm{GL}(n,F_p)$ is a finite field, performing at most $O(t{n}^6\log (p\left)\right)$ elementary operations in $F_p$. In the affirmative case, the algorithm returns a defining element a so that $F_p[A_1,\dots ,A_t]=F_p\left[a\right]$.

We then study an invariant for the extended-affine equivalence of Dembowski-Ostrom (DO) polynomials. More precisely, for a DO polynomial $g\in F_{p^n}\left[x\right]$, we associate to g a set of $n\times n$ matrices with coefficients in $F_p$, denoted $\mathrm{Quot}\left(D_g\right)$, 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, $\mathrm{Quot}\left(D_g\right)$ is the set of quotients $X{Y}^{-1}$ with $Y\ne 0,X$ being elements from the spread set of the corresponding commutative presemifield, and $\mathrm{Quot}\left(D_g\right)$ 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 $\mathrm{Quot}\left(D_g\right)$ 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 $X\in \mathrm{Quot}\left(D_g\right)$ generates a field $F_p\left[X\right]\subseteq \mathrm{Quot}\left(D_g\right)$ and $\mathrm{Quot}\left(D_g\right)$ contains the field $F_{p^n}$.