论文标题
在矩阵代数上，同构与有限的田地和平面dembowski-ostrom单位
On Matrix Algebras Isomorphic to Finite Fields and Planar Dembowski-Ostrom Monomials
论文作者
论文摘要
令$ p $为素数，$ n $是正整数。作为第一个主要结果，我们提出了一种确定算法，用于确定矩阵代数$ \ mathbb {f} _p [a_1，\ dots，a_t，a_t] $ at $ a_1，\ dots，a_t \ dots，a _t \ in \ mathrm {gl}（n，\ mathbbbb { $ \ MATHCAL {O}（tn^6 \ log（p））$基本操作$ \ Mathbb {f} _p $。在肯定的情况下，该算法返回一个定义元素$ a $，因此$ \ mathbb {f} _p [a_1，\ dots，a_t] = \ mathbb {f} _p [a] $。 然后，我们研究了Dembowski-Ostrom（DO）多项式的扩展植入等效性的不变性。更准确地说，对于多项式$ g \ in \ mathbb {f} _ {p^n} [x] $，我们将$ g $与$ n \ times n $矩阵与$ \ mathbbbb {f} _p $ in n in $ n \ times n $矩阵相关联将扩展的属于等效变换应用于$ g $时，矩阵相似性的不变性相似。如果$ g $是平面do多项式，则$ \ mathrm {quet}（\ mathcal {d} _g）$是$ xy^{ - 1} $的集合，$ xy^{ - 1} $，$ y \ neq 0，x $是相应的通勤式上的emperemifiemifielfieldield和$ \ mathrm {quot}（\ Mathcal {d} _g）$在$ p^n $的情况下形成$ p^n $的字段，并且仅当$ g $等于平面$ x^2 $，即当时仅当与$ g $相关的交换式Presemif与$ G $相关的情况与有限的字段相同。 作为第二个主要结果，我们分析了$ \ mathrm {quot}的结构（\ Mathcal {d} _g）$对于所有平面do do do no no no no no no no no no no no no no no no no nosotations of ofd off ofd ofd off ofd ofders ofd ofders of ods ofder ofders of ofder corders of odsotife to to to Pricite fort to Pricite of Picotopic as to Pricitite Field或换档扭曲场。更准确地说，对于$ g $等同于平面do noumalial，我们表明每个非零元素$ x \ in \ mathrm {quet}}（\ mathcal {d} _g）$生成一个field $ \ mathbb {f} _p {f} _p [X] $ \ mathrm {quet}（\ Mathcal {d} _g）$包含字段$ \ Mathbb {f} _ {p^n} $。
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 $\mathbb{F}_{p^n}$.
