The cycle structure of unicritical polynomials

TL;DR

This paper investigates the cycle structures of unicritical polynomial dynamical systems over finite fields, demonstrating that their cycle statistics are as random as possible and confirming a conjecture about the abundance of cycles.

Contribution

It proves that the cycle statistics of these polynomial families are maximally random and verifies a conjecture regarding the number of cycles in most cases.

Findings

Cycle structures are as random as possible

Most polynomials have many cycles

Addresses a conjecture by Mans et al.

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 $\mathbb{F}_p$. The questions of whether and in what sense these families are random have been studied extensively, spurred in part by Pollard's famous "rho" algorithm for integer factorization (the heuristic justification of which is the randomness of one such family). However, the cycle structure of these families cannot be random, since in any such family, the number of cycles of a fixed length in any dynamical system in the family is bounded. In this paper, we show that the cycle statistics of many of these families are as random as possible. As a corollary, we show that most members of these families have many cycles, addressing a conjecture of Mans et. al.