Settled polynomials over finite fields

Abstract

Abstract. We study the factorization into irreducibles of iterates of a quadratic polynomial f over a finite field. We call f settled when the factorization of its nth iterate for large n is dominated by "stable" polynomials, namely those that are irreducible under post-composition by any iterate of f . We prove that stable polynomials may be detected by their action on the critical orbit of f , and that the the critical orbit also gives information about the splitting of non-stable polynomials under post-composition by iterates of f . We then define a Markov process based on the critical orbit of f and conjecture that its limiting distribution describes the full factorization of large iterates of f . This conjecture implies that almost all quadratic f defined over a finite field are settled. We give several types of evidence for our conjecture.

On irreducible divisors of iterated polynomials

“…In [11] the critical orbit of quadratic polynomials f is defined as the set {f (n) (γ) | n 2} ∪ {−f (γ)}, where γ is the root of the derivative. This coincides with the following set…”

“…Following [1,2,10,11,15], we say that a polynomial f ∈ K[X] is stable if all iterates are irreducible over K.…”

On Stable Quadratic Polynomials

“…We need first the following result (see [13,Lemma 2.5]), which characterises completely the stability of quadratic polynomials over finite fields:…”

“…As in [13], for a quadratic polynomial f (X) = aX 2 + bX + c ∈ K [X], where the characteristic of K is not 2, we define γ = −b/2a as the unique critical point of f (that is, zero of the derivative f ) and consider the set If K = ‫ކ‬ q , q odd, clearly there is some t such that f (t) (γ ) = f (s) (γ ) for some positive integer s < t. Then f (n+t) (γ ) = f (n+s) (γ ) for any n 0. Accordingly, for the smallest value of t with the above property denoted by t f , we have Orb(f ) = {f (n) (γ ) : n = 2, .…”

“…It is shown in [11,12,13] that critical orbits play a very important role in the dynamics of polynomial iterations. In particular, by [13,Proposition 2.3], a quadratic polynomial f (X) ∈ K[X] is stable if the set {−f (γ )} ∪ Orb(f ) contains no squares. In the case when K = ‫ކ‬ q is a finite field of odd characteristic, this property is also necessary.…”