Arithmetic dynamics of random polynomials

TL;DR

This paper studies the statistical properties of polynomial dynamics over rationals, proving average bounds on rational preperiodic points and analyzing the behavior of dynamical minima, thus advancing understanding of conjectures in arithmetic dynamics.

Contribution

It establishes the average number of rational preperiodic points as zero and introduces the concept of dynamical successive minima, analyzing their average behavior.

Findings

Average number of rational preperiodic points is zero.

Proves an optimal statistical version of the dynamical Lang conjecture.

Studies the invariance and average behavior of dynamical successive minima.

Abstract

We investigate from a statistical perspective the arithmetic properties of the dynamics of polynomials of fixed degree and defined over the field of rational numbers. To start with, ordering their affine conjugacy classes by height, we show that their average number of rational preperiodic points is equal to zero, thereby proving a strong average version of the uniform boundedness conjecture of Morton and Silverman. Next, inspired by the analogy with the successive minima of a lattice we define the dynamical successive minima of a polynomial. Noting that these quantities are invariant under the action by conjugacy of the affine group we study their average behaviour using the aforementioned ordering by height. In particular, we prove an optimal statistical version of the dynamical Lang conjecture on the canonical height of rational non-preperiodic points.