Stability of certain higher degree polynomials

Abstract

One of the interesting problems in arithmetic dynamics is to study the stability of polynomials over a field. A polynomial [Formula: see text] is stable over [Formula: see text] if irreducibility of [Formula: see text] implies that all its iterates are also irreducible over [Formula: see text], that is, [Formula: see text] is irreducible over [Formula: see text] for all [Formula: see text], where [Formula: see text] denotes the [Formula: see text]-fold composition of [Formula: see text]. In this paper, we study the stability of [Formula: see text] for [Formula: see text], [Formula: see text]. We show that for infinite families of [Formula: see text], whenever [Formula: see text] is irreducible, all its iterates are irreducible, that is, [Formula: see text] is stable. Under the assumption of explicit [Formula: see text]-conjecture, we further prove the stability of [Formula: see text] for the remaining values of [Formula: see text]. Also for [Formula: see text], if [Formula: see text] is reducible, then the number of irreducible factors of each iterate of [Formula: see text] is exactly [Formula: see text] for [Formula: see text].