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

Angelot Behajaina; Elad Paran

doi:10.1016/j.ffa.2023.102265

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

Angelot Behajaina, Elad Paran

Mathematics discipline

Research output: Contribution to journal › Article › peer-review

Abstract

We study the analogue of the Collatz map in the polynomial ring F_p[x], for any prime number p, and the corresponding dynamical system. We show that every f∈F_p[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 F_p[[x]], observe that in F_p[[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 language	English
Article number	102265
Journal	Finite Fields and Their Applications
Volume	91
DOIs	https://doi.org/10.1016/j.ffa.2023.102265
State	Published - Oct 2023

Bibliographical note

Funding Information:
The first author is grateful for the support of the Israel Science Foundation (grant no. 353/21 ). We also thank the anonymous referee for his/her feedback and comments.

Publisher Copyright:
© 2023 Elsevier Inc.

Keywords

Collatz problem
Dynamics
Finite fields
Polynomials

Access to Document

10.1016/j.ffa.2023.102265

Cite this

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