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

  title={Irreducible compositions of degree two polynomials over finite fields have regular structure},
  author={Andrea Ferraguti and Giacomo Micheli and Reto Schnyder},
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. 

Figures from this paper

Dynamical height growth: left, right, and total orbits
Let $S$ be a set of dominant rational self-maps on $\mathbb{P}^N$. We study the arithmetic and dynamical degrees of infinite sequences of $S$ obtained by sequentially composing elements of $S$ on the
Full orbit sequences in affine spaces via fractional jumps and pseudorandom number generation
This paper provides a general theory to produce full orbit sequences in the affine $n$-dimensional space over a finite field for the case of the Inversive Congruential Generators.
A note on the factorization of iterated quadratics over finite fields
BSTRACT . f be a monic quadratic polynomial over a finite field of odd characteristic. In 2012, Boston and Jones constructed a Markov process based on the post-critical orbit of f , and conjectured
The set of stable primes for polynomial sequences with large Galois group
Let $K$ be a number field with ring of integers $\mathcal O_K$, and let $\{f_k\}_{k\in \mathbb N}\subseteq \mathcal O_K[x]$ be a sequence of monic polynomials such that for every $n\in \mathbb N$,
The inverse problem for arboreal Galois representations of index two
This paper introduces a systematic approach towards the inverse problem for arboreal Galois representations of finite index attached to quadratic polynomials. Let $F$ be a field of characteristic
Finite orbit points for sets of quadratic polynomials
  • W. Hindes
  • Mathematics
    International Journal of Number Theory
  • 2019
Let [Formula: see text] be a set of quadratic polynomials with rational coefficients, and let [Formula: see text] be a rational basepoint. We classify the pairs [Formula: see text] for which
The Algebraic Theory of Fractional Jumps
An efficient construction of a fractional jump of a projective map and the compound generator construction for the Inversive Congruential Generator is extended to Fractional jump sequences.
An equivariant isomorphism theorem for mod $\mathfrak {p}$ reductions of arboreal Galois representations
Let $\phi$ be a quadratic, monic polynomial with coefficients in $\mathcal O_{F,D}[t]$, where $\mathcal O_{F,D}$ is a localization of a number ring $\mathcal O_F$. In this paper, we first prove that
Current trends and open problems in arithmetic dynamics
Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in


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
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.
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