Prescribing digits in finite fields
type: Item
instance of: scholarly article
title: Prescribing digits in finite fields
zbMATH Open document ID: 1448.11215
author: Cathy Swaenepoel
published in: Journal of Number Theory
publication date: 2018-05-02T00:00:00Z
review text: Consider the finite field \(\mathbb{F}_q\), where \(q=p^r\) with \(r\geq 2\) and \(p\) a prime number, and fix a basis \(\{e_1,\ldots, e_r\}\) of the vector space \(\mathbb{F}_q\) over \(\mathbb{F}_p\). Since every \(x\in \mathbb{F}_q\) has a unique representation \(x=\sum_{j=1}^r c_j e_j\) with \(c_1,\ldots, c_r\in \mathbb{F}_p\), then we may call \(c_j\) the \(j\)-th digit of \(x\). Results here deal with estimates on the number of field elements given that exactly \(k\) of the \(r\) digits are prescribed. More precisely, let \(P\in \mathbb{F}_q[X]\) be a polynomial of degree \(n\), where \(p\nmid n\), and let \(J\subseteq \{1,\ldots, r\}\) with \(|J|=k\) and \(\alpha=(\alpha_j)_{j\in J}\in (\mathbb{F}_p)^k\). Define \(\mathcal{F}_q(P,k,J,\alpha)\) to be the set of all \(x\in \mathbb{F}_q\) such that the \(j\)-th digit of \(P(x)\) equals \(\alpha_j\) for each \(j\in J\). Then (Theorem 1.1) \[ \left| \; |\mathcal{F}_q(P,k,J,\alpha)| - \frac{q}{p^k}\right| \leq \frac{p^{k}-1}{p^k}\, (n-1)\, \sqrt q\; . \] In particular (Corollary 1.2) for up to half of prescribed digits, an asymptotic result is obtained as \(p^r\to\infty\), where \[ |\mathcal{F}_{p^r}(P,k,J,\alpha)| = p^{r-k} \, (1+o(1)) \] uniformly over \(k\leq (\frac{1}{2}-\varepsilon) r\) for any \(\varepsilon >0\). Both Theorem 1.1 and Corollary 1.2 are developed further for the particular case \(P=aX^2\) (Theorem 1.3 and Corollaries 1.4, 1.5) as well as the case in which \(x\) is considered only among the primitive elements of \(\mathbb{F}_q^*\) (Theorem 1.6 and Corollary 1.8).
reviewed by: Amin Witno
Mathematics Subject Classification ID: 11T23
Mathematics Subject Classification ID: 11A63
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 6865879
zbMATH Keywords: finite fields
zbMATH Keywords: prescribed digits
zbMATH Keywords: character sums
zbMATH Keywords: squares
zbMATH Keywords: primitive elements
MaRDI profile type: Publication
full work available at URL: j.jnt.2017.11.012
OpenAlex ID: W2776247991
cites work: Prescribing the binary digits of primes
cites work: Prescribing the binary digits of primes. II
cites work: Polynomial values and generators with missing digits in finite fields
cites work: The sum of digits function in finite fields
cites work: Sums of digits of multiples of integers
cites work: CONGRUENCES DE SOMMES DE CHIFFRES DE VALEURS POLYNOMIALES
cites work: Prescribing the binary digits of squarefree numbers and quadratic residues
cites work: On arithmetic properties of integers with missing digits. I: Distribution in residue classes
cites work: On arithmetic properties of integers with missing digits. II: Prime factors
cites work: Sur les nombres qui ont des propriétés additives et multiplicatives données
cites work: Q3856819
cites work: Primes with preassigned digits
cites work: Primes with preassigned digits II
cites work: The sum of digits of squares
cites work: On a problem of Gelfond: the sum of digits of prime numbers
cites work: Primes with restricted digits
cites work: Handbook of finite fields
cites work: Equations over finite fields. An elementary approach
cites work: Sur les nombres premiers ayant des chiffres initiaux et finals donnés
DOI: 10.1016/J.JNT.2017.11.012
description: scientific article; zbMATH DE number 6865879