Stability of certain higher degree polynomials

Article Type

Research Article

Publication Title

International Journal of Number Theory

Abstract

One of the interesting problems in arithmetic dynamics is to study the stability of polynomials over a field. A polynomial f(z) ∈ Q[z] is stable over Q if irreducibility of f(z) implies that all its iterates are also irreducible over Q, that is, fn(z) is irreducible over Q for all n ≥ 1, where fn(z) denotes the n-fold composition of f(z). In this paper, we study the stability of f(z) = zd + 1/c for d ≥ 2, c ∈ Z\{0}. We show that for infinite families of d ≥ 3, whenever f(z) is irreducible, all its iterates are irreducible, that is, f(z) is stable. Under the assumption of explicit abc-conjecture, we further prove the stability of f(z) = zd + 1/c for the remaining values of d. Also for d = 3, if f(z) is reducible, then the number of irreducible factors of each iterate of f(z) is exactly 2 for |c| ≤ 1012. 1012.

First Page

229

Last Page

240

DOI

10.1142/S1793042124500118

Publication Date

2-1-2024

Comments

Open Access; Green Open Access

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