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

@article{Ferraguti2017IrreducibleCO,
  title={Irreducible compositions of degree two polynomials over finite fields have regular structure},
  author={Andrea Ferraguti and Giacomo Micheli and Reto Schnyder},
  journal={ArXiv},
  year={2017},
  volume={abs/1701.06040}
}
Let $q$ be an odd prime power and $D$ be the set of monic irreducible polynomials in $\mathbb F_q[x]$ which can be written as a composition of monic degree two polynomials. In this paper we prove that $D$ has a natural regular structure by showing that there exists a finite automaton having $D$ as accepted language. Our method is constructive. 

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