论文标题
$ r $ - $ k $ - 正常元素在有限领域的算术进展中
$r$-primitive $k$-normal elements in arithmetic progressions over finite fields
论文作者
论文摘要
令$ \ mathbb {f} _ {q^n} $是一个有限字段，具有$ q^n $元素。对于$ q^n-1 $的积极除数$ r $，如果其多重订单为$（q^n-1）/r $，则称为\ mathbb {f} _ {q^n}^*$的元素$α\。另外，对于非阴性整数$ k $，元素$α\ in \ mathbb {f} _ {q^n} $是\ textit {$ k $ -normal}，$ \ mathbb {f} _q $ if $ \ mathbb {f} _q $，如果α^{q^{n-2}} x +α^{q^{n-1}}，x^n-1）$ in $ \ mathbb {f} _ {q^n} [q^n} [x] $具有$ k $。 In this paper we discuss the existence of elements in arithmetic progressions $\{α, α+β, α+2β, \ldotsα+(m-1)β\} \subset \mathbb{F}_{q^n}$ with $α+(i-1)β$ being $r_i$-primitive and at least one of the elements in the arithmetic progression being $ k $ - normal of $ \ mathbb {f} _q $。当$ k = r_i = r_i = 2 $ for $ i \ in \ in \ {1，\ dots，m \} $时，我们会获得一般$ k，r_1，\ dots，r_m $和混凝土结果的一般渐近结果。
Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements. For a positive divisor $r$ of $q^n-1$, the element $α\in \mathbb{F}_{q^n}^*$ is called \textit{$r$-primitive} if its multiplicative order is $(q^n-1)/r$. Also, for a non-negative integer $k$, the element $α\in \mathbb{F}_{q^n}$ is \textit{$k$-normal} over $\mathbb{F}_q$ if $\gcd(αx^{n-1}+ α^q x^{n-2} + \ldots + α^{q^{n-2}}x + α^{q^{n-1}} , x^n-1)$ in $\mathbb{F}_{q^n}[x]$ has degree $k$. In this paper we discuss the existence of elements in arithmetic progressions $\{α, α+β, α+2β, \ldotsα+(m-1)β\} \subset \mathbb{F}_{q^n}$ with $α+(i-1)β$ being $r_i$-primitive and at least one of the elements in the arithmetic progression being $k$-normal over $\mathbb{F}_q$. We obtain asymptotic results for general $k, r_1, \dots, r_m$ and concrete results when $k = r_i = 2$ for $i \in \{1, \dots, m\}$.
