Dynamical Irreducibility of Certain Families of Polynomials over Finite Fields

TL;DR

This paper establishes precise conditions under which certain unicritical polynomials and other polynomial families are dynamically irreducible over finite fields, extending previous results and exploring new polynomial classes.

Contribution

It provides necessary and sufficient criteria for dynamical irreducibility of unicritical, cubic, and shifted linearized polynomials over finite fields, broadening existing knowledge.

Findings

Criteria for unicritical polynomial irreducibility over finite fields

Extension of previous results by Boston-Jones and Hamblen-Jones-Madhu

Analysis of cubic and shifted linearized polynomial irreducibility

Abstract

We determine necessary and sufficient conditions for unicritical polynomials to be dynamically irreducible over finite fields. This result extends the results of Boston-Jones and Hamblen-Jones-Madhu regarding the dynamical irreducibility of particular families of unicritical polynomials. We also investigate dynamical irreducibility conditions for cubic and shifted linearized polynomials.