The Collatz problem in Fp[x] and Fp[[x]]

Angelot Behajaina, Elad Paran

Research output: Contribution to journalArticlepeer-review

Abstract

We study the analogue of the Collatz map in the polynomial ring Fp[x], for any prime number p, and the corresponding dynamical system. We show that every f∈Fp[x] is eventually periodic in this system, in a quadratic number of iterations in deg(f), and describe explicitly all corresponding cycles. This extends a result of Hicks, Mullen, Yucas and Zavislak, who studied the case p=2. We also study the Collatz map in the formal power series ring Fp[[x]], observe that in Fp[[x]] all but countably many power series generate divergent trajectories via iterations of this map, and characterize those power series that are eventually periodic.

Original languageEnglish
Article number102265
JournalFinite Fields and Their Applications
Volume91
DOIs
StatePublished - Oct 2023

Bibliographical note

Publisher Copyright:
© 2023 Elsevier Inc.

Keywords

  • Collatz problem
  • Dynamics
  • Finite fields
  • Polynomials

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