On enumeration of irreducible polynomials and related objects over a finite field with respect to their trace and norm
type: Item
instance of: scholarly article
title: On enumeration of irreducible polynomials and related objects over a finite field with respect to their trace and norm
zbMATH Open document ID: 1500.11087
author: Simon Kuttner
author: Zhicheng Gao
author: Qiang Wang
published in: Finite Fields and their Applications
publication date: 2021-02-19T00:00:00Z
review text: Let \(\mathbb F_q\) be the finite field with \(q\) elements and let \(p=\text{char }\mathbb F_q\). For \(\gamma_1\in\mathbb F_q\), \(\gamma_2\in\mathbb F_q^*\) and \(m>0\), let \(I_m(\gamma_1,\gamma_2)\) denote the number of irreducible polynomials \(f(x)=x^m-a_{m-1}x^{m-1}+\cdots+(-1)^ma_0\in\mathbb F_q[x]\) with \(a_{m-1}=\gamma_1\) and \(a_0=\gamma_2\). The asymptotic formula for \(I_m(\gamma_1,\gamma_2)\), as \(m\to\infty\) and/or \(q\to\infty\), was a topic studied by many authors. The main term in the asymptotic formula for \(I_m(\gamma_1,\gamma_2)\) is known to be \(q^{m-1}/m(q-1)\). The objective of the present paper is to improve the previous bounds for the error term in the asymptotic formula for \(I_m(\gamma_1,\gamma_2)\). It is known that \[ I_m(\gamma_1,\gamma_2)=\frac 1m\sum_{d\mid m}\mu(m/d)\sum_{\alpha,\beta\in\mathbb F_q}N_d(\alpha,\beta)[\![(m/d)\alpha=\gamma_1,\,\beta^{m/d}=\gamma_2]\!], \] where \(\mu\) is the Möbius function, \[ N_d(\alpha,\beta)=|\{u\in\mathbb F_{q^d}:\text{trace}_{q^d/q}(\alpha)=\gamma_1,\; \text{norm}_{q^d/q}(\alpha)=\gamma_2\}|, \] and \[ [\![P]\!]=\begin{cases} 1&\text{if property \(P\) is true},\cr 0&\text{otherwise}. \end{cases} \] Moreover, \[ N_d(\gamma_1,\gamma_2)=\frac{q^{d-1}-1}{q-1}+(-1)^{d-1}\Bigl(M_d(\gamma_1,\gamma_2)-\frac{(q-1)^{d-1}-(-1)^{d-1}}q\Bigr), \] where \[ M_d(\gamma_1,\gamma_2)=\Bigl|\Bigl\{(\alpha_1,\dots,\alpha_d)\in{\mathbb F_q^*}^d:\sum_{j=1}^d\alpha_j=\gamma_1,\ \prod_{j=1}^d\alpha_j=\gamma_2\Bigr\}\Bigr|. \] A proof of the formula for \(N_d(\gamma_1,\gamma_2)\) using generating functions is included in the paper. The number \(M_d(\gamma_1,\gamma_2)\) can be expressed in terms of Gauss sums over \(\mathbb F_q\). The estimates for \(I_m(\gamma_1,\gamma_2)\) are obtained from the relation between \(I_m(\gamma_1,\gamma_2)\) and \(M_d(\gamma_1,\gamma_2)\) and the basic properties of Gauss sums. The main result (Theorem 2) gives two upper bounds for \(|I_m(0,\gamma_2)-A|\) and \(|I_m(\gamma_1,\gamma_2…
Mathematics Subject Classification ID: 11T06
Mathematics Subject Classification ID: 12E05
zbMATH DE Number: 7313134
zbMATH Keywords: irreducible polynomials
zbMATH Keywords: trace
zbMATH Keywords: norm
zbMATH Keywords: Gauss sum
zbMATH Keywords: mean
zbMATH Keywords: variance
MaRDI profile type: Publication
full work available at URL: j.ffa.2020.101770
OpenAlex ID: W3096742162
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DOI: 10.1016/J.FFA.2020.101770
description: scientific article; zbMATH DE number 7313134