An Infinite Family of Cubics with Emergent Reducibility at Depth 1

Abstract

A polynomial $f\left(x\right)$ has emergent reducibility at depth $n$ if $f^{\circ k}\left(x\right)$ is irreducible for $0\le k\le n-1$ but $f^{\circ n}\left(x\right)$ is reducible. In this paper we prove that there are infinitely many irreducible cubics $f\in Z\left[x\right]$ with $f\circ f$ reducible by exhibiting a one parameter family with this property.