计数具有限制系数的不可约多项式

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications Pub Date : 2025-07-07 DOI:10.1016/j.ffa.2025.102691

Kaimin Cheng

引用次数: 0

摘要

设q是一个素数幂，设Fq表示有q个元素的有限域。考虑一个正整数n，设R={Ri}i=0n−1是Fq的子集族。定义N（R， N）为N / Fq次的不可约一元多项式的个数，其中每个非前导项Ti的系数在Fq∈Ri中。本文给出了N（R， N）的渐近公式，将Porritt的结果推广到更一般的情况。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Counting irreducible polynomials with restricted coefficients

Let q be a prime power, and let

F_{q}

denote the finite field with q elements. Consider a positive integer n, and let

R = {R_{i}}_{i = 0}^{n - 1}

be a family of subsets of

F_{q}

. Define

N (R, n)

as the number of monic irreducible polynomials of degree n over

F_{q}

where the coefficient of each non-leading term

T^{i}

lies in

F_{q} ∖ R_{i}

. In this paper, we provide an asymptotic formula for

N (R, n)

, extending a result of Porritt to a more general case.

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来源期刊

Finite Fields and Their Applications 数学-数学

CiteScore

2.00

自引率

20.00%

发文量

133

审稿时长

6-12 weeks

期刊介绍： Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering. For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods. The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.