On sets of irreducible polynomials closed by composition

TL;DR

This paper characterizes when a semigroup generated by degree 2 polynomials over finite fields consists solely of irreducible polynomials, using a graph-based criterion that links algebraic properties to combinatorial structures.

Contribution

It provides a necessary and sufficient condition for such semigroups to contain only irreducible polynomials, based on a graph determined by the generating set.

Findings

Characterization of irreducible polynomial semigroups via graph conditions

Examples of semigroups generated by two degree 2 polynomials

Non-existence results for certain sets over infinite prime fields

Abstract

Let $\mathcal S$ be a set of monic degree $2$ polynomials over a finite field and let $C$ be the compositional semigroup generated by $\mathcal S$. In this paper we establish a necessary and sufficient condition for $C$ to be consisting entirely of irreducible polynomials. The condition we deduce depends on the finite data encoded in a certain graph uniquely determined by the generating set $\mathcal S$. Using this machinery we are able both to show examples of semigroups of irreducible polynomials generated by two degree $2$ polynomials and to give some non-existence results for some of these sets in infinitely many prime fields satisfying certain arithmetic conditions.