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

TL;DR

This paper presents a deterministic algorithm to identify when matrix algebras over finite fields are themselves finite fields and explores invariants of Dembowski-Ostrom polynomials, linking algebraic structures to finite field properties.

Contribution

It introduces a new algorithm for recognizing finite field structures within matrix algebras and analyzes invariants of DO polynomials related to finite field isomorphisms and semifield structures.

Findings

Algorithm decides if matrix algebra is a finite field in polynomial time.

Invariant set $ ext{Quot}( ext{D}_g)$ characterizes extended-affine equivalence of DO polynomials.

$ ext{Quot}( ext{D}_g)$ forms a field of order $p^n$ iff $g$ is equivalent to $x^2$.

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 $\mathbb{F}_p[A_1,\dots,A_t]$ 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 $\mathbb{F}_{p^n}$.