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.
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Acknowledgements
The first author was supported by the Swiss National Science Foundation grant number 168459. The second author was supported by Swiss National Science Foundation grant number 161757. The third author was supported in part by Swiss National Science Foundation grant number 149716 and Armasuisse.
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Ferraguti, A., Micheli, G., Schnyder, R. (2016). On Sets of Irreducible Polynomials Closed by Composition. In: Duquesne, S., Petkova-Nikova, S. (eds) Arithmetic of Finite Fields. WAIFI 2016. Lecture Notes in Computer Science(), vol 10064. Springer, Cham. https://doi.org/10.1007/978-3-319-55227-9_6
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