The Collatz map analogue in polynomial rings and in completions

Angelot Behajaina, Elad Paran

Research output: Contribution to journalArticlepeer-review

Abstract

We study an analogue of the Collatz map in the polynomial ring R[x], where R is an arbitrary commutative ring. We prove that if R is of positive characteristic, then every polynomial in R[x] is eventually periodic with respect to this map. This extends previous works of the authors and of Hicks, Mullen, Yucas and Zavislak, who studied the Collatz map on Fp[x] and F2[x], respectively. We also consider the Collatz map on the ring of formal power series R[[x]] when R is finite: we characterize the eventually periodic series in this ring, and give formulas for the number of cycles induced by the Collatz map, of any given length. We provide similar formulas for the original Collatz map defined on the ring Z2 of 2-adic integers, extending previous results of Lagarias.

Original languageEnglish
Article number114273
JournalDiscrete Mathematics
Volume348
Issue number1
DOIs
StatePublished - Jan 2025

Bibliographical note

Publisher Copyright:
© 2024 Elsevier B.V.

Keywords

  • Collatz map
  • Polynomials

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