An infinite family of cubics with emergent reduciability at depth 1

Abstract

A polynomial ƒ(x) has emergent reducibility at depth n if ƒ ^ok(x) is irreducible for 0≤ k ≤ n - 1 but ƒ ^on(x) is reducible. In this paper we prove that there are innitely many irreducible cubics ƒ ∈ Ζ[x] with ƒ o ƒ reducible by exhibiting a one parameter family with this property.

Mathematics Subject Classication (2010): Primary 11R09; Secondary 37P05, 11D25.

Key words: Cubics, iterates, arithmetic dynamics.