Eventual Stability of pure polynomials over the rational field

TL;DR

This paper proves that pure polynomials over the rationals become stable in their factorization patterns upon iteration, introduces new classes of irreducible polynomials, and characterizes polynomials with pure iterates.

Contribution

It establishes the eventual stability of pure polynomials over rationals and introduces methods for generating irreducible polynomials through iteration.

Findings

Pure polynomials are eventually stable with bounded irreducible factors.

Dumas polynomials are dynamically irreducible under certain conditions.

Characterization of polynomials with non-pure degrees but pure iterates.

Abstract

A polynomial with rational coefficients is said to be pure with respect to a rational prime $p$ if its Newton polygon has one slope. In this article, we prove that the number of irreducible factors of the $n$-th iterate of a pure polynomial over the rational field $\mathbb Q$ is bounded independent of $n$. In other words, we show that pure polynomials are {\em eventually stable}. Consequently, several eventual stability results available in literature follow; including the eventual stability of the polynomial $x^d+c\in\mathbb{Q}[x]$, where $c\ne 0,1$, is not a reciprocal of an integer. In addition, we establish the dynamical irreducibility, i.e., the irreducibility of all iterates, of a subfamily of pure polynomials, namely Dumas polynomials with respect to a rational prime $p$ under a mild condition on the degree. This provides iterative techniques to produce irreducible polynomials in $\mathbb{Q}[x]$ by composing pure polynomials of different degrees. During the course of this work, we characterize all polynomials whose degrees are large enough that are not pure, yet they possess pure iterates. This implies the existence of polynomials whose shifts are all dynamically irreducible in $\mathbb{Z}[x]$.