On Sets of Irreducible Polynomials Closed by Composition

Abstract

Abstract. Let S be a set of monic degree 2 polynomials over a finite field and let C be the compositional semigroup generated by 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 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.

On the complexity of exact counting of dynamically irreducible polynomials

“…In the following we extend some results of Ferraguti, Micheli and Schnyder [4] and Heath-Brown and Micheli [8] about dynamically irreducible sets for non-monic quadratic polynomials.…”

“…Ferraguti, Micheli and Schnyder [4] have characterized the sets of monic quadratic polynomial to be dynamically irreducible in terms of the unique critical points of the polynomials. We also note that the subsequent work [5] gives a representation of the set dynamically irreducible polynomials via finite automata.…”

Constraining Images of Quadratic Arboreal Representations

“…This is still a rather mysterious topic even in the first non-trivial case, which is that of quadratic polynomials. A great amount of interest on the topic has risen in recent years, as witnessed by the numerous papers on the topic, such as [1,3,4,7,9,10,11,12,13,20,21].…”

Irreducible compositions of degree two polynomials over finite fields have regular structure

“…It has been of great interest in recent years the study of irreducible polynomials which can be written as composition of degree two polynomials (see for example [1,2,6,8,9,11,13,14,16,17,18]). Such polynomials are also used in other contexts, see for example Rafe Jones' construction of irreducible polynomials reducible modulo every prime [16] or the proof of [3, Conjecture 1.2] in [5].…”