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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References

SHOWING 1-10 OF 19 REFERENCES
Irreducible polynomials over finite fields produced by composition of quadratics
For a set $S$ of quadratic polynomials over a finite field, let $C$ be the (infinite) set of arbitrary compositions of elements in $S$. In this paper we show that there are examples with arbitrarily…
On Sets of Irreducible Polynomials Closed by Composition
TLDR
This paper establishes a necessary and sufficient condition for C to be consisting entirely of irreducible polynomials, which depends on the finite data encoded in a certain graph uniquely determined by the generating set.
ON STABLE QUADRATIC POLYNOMIALS
Abstract We recall that a polynomial f(X) ∈ K[X] over a field K is called stable if all its iterates are irreducible over K. We show that almost all monic quadratic polynomials f(X) ∈ ℤ[X] are stable…
An iterative construction of irreducible polynomials reducible modulo every prime
Probabilistic Algorithms in Finite Fields
  • M. Rabin
  • Computer Science, Mathematics
    SIAM J. Comput.
  • 1980
We present probabilistic algorithms for the problems of finding an irreducible polynomial of degree n over a finite field, finding roots of a polynomial, and factoring a polynomial into its…
On the length of critical orbits of stable quadratic polynomials
We use the Weil bound of multiplicative character sums together with some recent results of N. Boston and R. Jones, to show that the critical orbit of quadratic polynomials over a finite field of q…
Settled polynomials over finite fields
We study the factorization into irreducibles of iterates of a qua- dratic polynomial f over a nite eld. We call f settled when the factorization of its nth iterate for large n is dominated by…
An estimate on the number of stable quadratic polynomials
Functional Decomposition of Polynomials: The Tame Case
  • J. Gathen
  • Mathematics, Computer Science
    J. Symb. Comput.
  • 1990
A Note on Stable Quadratic Polynomials over Fields of Characteristic Two
In this note, first we show that there is no stable quadratic polynomial over finite fields of characteristic two and then show that there exist stable quadratic polynomials over function fields of…
...
1
2
...