TY - JOUR

T1 - The Collatz map analogue in polynomial rings and in completions

AU - Behajaina, Angelot

AU - Paran, Elad

N1 - Publisher Copyright:
© 2024 Elsevier B.V.

PY - 2025/1

Y1 - 2025/1

N2 - 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.

AB - 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.

KW - Collatz map

KW - Polynomials

UR - http://www.scopus.com/inward/record.url?scp=85205352599&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2024.114273

DO - 10.1016/j.disc.2024.114273

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85205352599

SN - 0012-365X

VL - 348

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 1

M1 - 114273

ER -