Groups generated by iterations of polynomials over finite fields

Igor E. Shparlinski*

*Corresponding author for this work

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

Given a finite field of Fq elements, we consider a trajectory of the map u→ f(u) associated with a polynomial f ∈ Fq[X]. Using bounds of character sums, under some mild condition on f, we show that for an appropriate constant C > 0 no N ≥ Cq1/2 distinct consecutive elements of such a trajectory are contained in a small subgroup G of F∗q, improving the trivial lower bound #G ≥ N. Using a different technique, we also obtain a similar result for very small values of N. These results are multiplicative analogues of several recently obtained bounds on the length of intervals containing N distinct consecutive elements of such a trajectory.

Original languageEnglish
Pages (from-to)235-245
Number of pages11
JournalProceedings of the Edinburgh Mathematical Society
Volume59
Issue number1
DOIs
Publication statusPublished - Feb 2016

Keywords

  • finite fields
  • polynomial maps
  • subgroups
  • character sums
  • POINTS
  • MAPS

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