An Infinite Family of Cubics with Emergent Reducibility at Depth 1

TL;DR

This paper demonstrates the existence of infinitely many cubic polynomials with a specific reducibility pattern, where the polynomial remains irreducible under iteration until the first iteration, at which point it becomes reducible.

Contribution

The paper introduces a one-parameter family of cubic polynomials over integers that exhibit emergent reducibility at depth one, establishing an infinite family with this property.

Findings

Infinite family of cubics with emergent reducibility at depth 1

Explicit construction of such polynomials

Proof of their irreducibility until the first iteration

Abstract

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