Polynomial values and generators with missing digits in finite fields

Cécile Dartyge, Christian Mauduit, András Sárközy

Research output: Contribution to journalArticle

7 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 = Σr j=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

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

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