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 Sets of Irreducible Polynomials Closed by Composition

Ferraguti, Andrea;
2017-01-01

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.
Settore MAT/03 - Geometria
Arithmetic of Finite Fields. WAIFI 2016.
Springer
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11384/101125
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