Dynamically distinguishing polynomials

Abstract

A polynomial with integer coefficients yields a family of dynamical systems indexed by primes as follows: For any prime p, reduce its coefficients mod p and consider its action on the field F p . We say a subset of Z[x] is dynamically distinguishable mod p if the associated mod p dynamical systems are pairwise non-isomorphic. For any k, M ∈ Z >1 , we prove that there are infinitely many sets of integers M of size M such that x k + m | m ∈ M is dynamically distinguishable mod p for most p (in the sense of natural density). Our proof uses the Galois theory of dynatomic polynomials largely developed by Morton, who proved that the Galois groups of these polynomials are often isomorphic to a particular family of wreath products. In the course of proving our result, we generalize Morton's work and compute statistics of these wreath products.

The Cycle Structure of Unicritical Polynomials

“…Then, Section 3.2 computes useful bounds on ω n,r . These bounds generalize Theorem 3.5 of [BG17]; in Section 4.2 we apply these bounds to prove Theorem 4.7.…”

“…In Theorem 3.3, we compute all their moments. Then, in Theorem 3.6, we prove precise estimates for them; this theorem generalizes Theorem 3.5 of [BG17], which estimates only ω n,r (0). In Section 4.1 we prove the aforementioned Theorem 4.4, obtaining Corollary 1.2 as an immediate consequence.…”

“…The first result addresses the following inconvenience: for any f ∈ Z[x], n ∈ Z ≥1 , and p ∈ P, the polynomial [Φ f,n ] p certainly vanishes on the period n points of (F p , [f ] p ), but, as discussed in [BG17], it occasionally vanishes at points of lower period as well (indeed this can happen even in characteristic 0-see [Sil07, Example 4.2]). Luckily, as long as f n (x) − x has distinct roots, this pathology can only occur for finitely many p ∈ P. We record this fact as Proposition 4.1.…”

Periodic points of polynomials over finite fields

“…As an example of the utility of this approach, we recall that Pollard's famous "rho" method of factorization [Pol75] relies on an aspect of the purported randomness of the family of dynamical systems associated to R = Z and f = X 2 + 1. For recent work on this approach to studying randomness, see [JKMT16,BG17,BG20,Juu]. To ease notation, for any commutative ring R, we write P R for the nonzero prime ideals of R; moreover, if R is residually finite, then for any p ∈ P R , we write N (p) for [R] p .…”

Iteration entropy

“…Beginning with Flajolet and Odlyzko [1990], also Flynn and Garton [2014], Bellah et al [2016], Bridy and Garton [2017] studied functions and polynomials from this perspective. For a uniformly random map on n points, the expected size of the giant component (an undirected component of largest size) in its functional graph is µn with µ ≈ 0.75788 (Flajolet and Odlyzko [1990], Theorem 8 (ii)).…”