Arithmetic dynamics of a discrete Painlev\'e equation

TL;DR

This paper studies the behavior of discrete Painlevé equations over finite fields, revealing that their orbits are constrained to low-genus algebraic curves and satisfy bounds similar to classical number theory results.

Contribution

It explicitly characterizes the algebraic curves associated with these orbits and establishes bounds on their point counts, contrasting with complex solutions.

Findings

Orbits lie on algebraic curves with genus ≤ 1

Number of points in orbits satisfies the Hasse bound

Curves are explicitly given by defining polynomials

Abstract

We consider the orbits of a discrete Painlev\'e equation over finite fields and show that the number of points in such orbits satisfy the Hasse bound. The orbits turn out to lie on algebraic curves, whose defining polynomials are given explicitly. Moreover, these curves are shown to have genus less than or equal to one, which contrasts sharply with the case of discrete Painlev\'e equations over $\mathbb{C}$, whose generic solutions are believed to be more transcendental than elliptic functions.