Polynomial values and generators with missing digits in finite fields

Cécile Dartyge, Christian Mauduit, A. Sárközy

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

We consider the linear vector space formed by the elements of the finite field Fq with q = pr over Fp. Then the elements x of Fq have a unique representation in the form x = Σrj=1 cjaj with cj ∈ Fp; the coefficients cj will be called digits. Let D be a subset of Fp with 2 ≤ |D| <p. We consider elements x of Fq such that for their every digit cj we have cj ∈ D; then we say that the elements of Fp \ D are "missing digits". We will show that if D is a large enough subset of Fp, then there are squares with missing digits in Fq; if the degree of the polynomial f(x) ∈ Fq[X] is at least 2 then it assumes values with missing digits; there are generators g in Fq such that f(g) is of missing digits.

Original languageEnglish
Pages (from-to)65-74
Number of pages10
JournalFunctiones et Approximatio, Commentarii Mathematici
Volume52
Issue number1
DOIs
Publication statusPublished - Jul 1 2015

Fingerprint

Digit
Galois field
Generator
Polynomial
Subset
Vector space
Coefficient

Keywords

  • Character sums
  • Digits properties
  • Finite fields
  • Generators
  • Polynomials
  • Primitive roots
  • Squares

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Polynomial values and generators with missing digits in finite fields. / Dartyge, Cécile; Mauduit, Christian; Sárközy, A.

In: Functiones et Approximatio, Commentarii Mathematici, Vol. 52, No. 1, 01.07.2015, p. 65-74.

Research output: Contribution to journalArticle

@article{7612c1d25dec4495943ce09a37896ef6,
title = "Polynomial values and generators with missing digits in finite fields",
abstract = "We consider the linear vector space formed by the elements of the finite field Fq with q = pr over Fp. Then the elements x of Fq have a unique representation in the form x = Σrj=1 cjaj with cj ∈ Fp; the coefficients cj will be called digits. Let D be a subset of Fp with 2 ≤ |D| q such that for their every digit cj we have cj ∈ D; then we say that the elements of Fp \ D are {"}missing digits{"}. We will show that if D is a large enough subset of Fp, then there are squares with missing digits in Fq; if the degree of the polynomial f(x) ∈ Fq[X] is at least 2 then it assumes values with missing digits; there are generators g in Fq such that f(g) is of missing digits.",
keywords = "Character sums, Digits properties, Finite fields, Generators, Polynomials, Primitive roots, Squares",
author = "C{\'e}cile Dartyge and Christian Mauduit and A. S{\'a}rk{\"o}zy",
year = "2015",
month = "7",
day = "1",
doi = "10.7169/facm/2015.52.1.5",
language = "English",
volume = "52",
pages = "65--74",
journal = "Functiones et Approximatio, Commentarii Mathematici",
issn = "0208-6573",
publisher = "Adam Mickiewicz University Press",
number = "1",

}

TY - JOUR

T1 - Polynomial values and generators with missing digits in finite fields

AU - Dartyge, Cécile

AU - Mauduit, Christian

AU - Sárközy, A.

PY - 2015/7/1

Y1 - 2015/7/1

N2 - We consider the linear vector space formed by the elements of the finite field Fq with q = pr over Fp. Then the elements x of Fq have a unique representation in the form x = Σrj=1 cjaj with cj ∈ Fp; the coefficients cj will be called digits. Let D be a subset of Fp with 2 ≤ |D| q such that for their every digit cj we have cj ∈ D; then we say that the elements of Fp \ D are "missing digits". We will show that if D is a large enough subset of Fp, then there are squares with missing digits in Fq; if the degree of the polynomial f(x) ∈ Fq[X] is at least 2 then it assumes values with missing digits; there are generators g in Fq such that f(g) is of missing digits.

AB - We consider the linear vector space formed by the elements of the finite field Fq with q = pr over Fp. Then the elements x of Fq have a unique representation in the form x = Σrj=1 cjaj with cj ∈ Fp; the coefficients cj will be called digits. Let D be a subset of Fp with 2 ≤ |D| q such that for their every digit cj we have cj ∈ D; then we say that the elements of Fp \ D are "missing digits". We will show that if D is a large enough subset of Fp, then there are squares with missing digits in Fq; if the degree of the polynomial f(x) ∈ Fq[X] is at least 2 then it assumes values with missing digits; there are generators g in Fq such that f(g) is of missing digits.

KW - Character sums

KW - Digits properties

KW - Finite fields

KW - Generators

KW - Polynomials

KW - Primitive roots

KW - Squares

UR - http://www.scopus.com/inward/record.url?scp=84934344095&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84934344095&partnerID=8YFLogxK

U2 - 10.7169/facm/2015.52.1.5

DO - 10.7169/facm/2015.52.1.5

M3 - Article

VL - 52

SP - 65

EP - 74

JO - Functiones et Approximatio, Commentarii Mathematici

JF - Functiones et Approximatio, Commentarii Mathematici

SN - 0208-6573

IS - 1

ER -