On sets of irreducible polynomials closed by composition

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.

References in corpus (1)

• On the existence of infinite, non-trivial $F$-sets

Cited by in corpus (4)

• Irreducible compositions of degree two polynomials over finite fields have regular structure
• Irreducible polynomials over finite fields produced by composition of quadratics
• Full Orbit Sequences in Affine Spaces via Fractional Jumps and Pseudorandom Number Generation
• Fractional jumps: complete characterisation and an explicit infinite family