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
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AN - SCOPUS:85205352599
SN - 0012-365X
VL - 348
JO - Discrete Mathematics
JF - Discrete Mathematics
IS - 1
M1 - 114273
ER -