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Arithmetic dynamics of a discrete Painlevé equation

Nalini Joshi, Pieter Roffelsen

TL;DR

The work investigates the dynamics of the discrete Painlevé equation $q\textrm{P}_{\textrm{I}}$ over finite fields, showing that orbit lengths satisfy a Hasse-type bound and that orbits reside on explicitly described algebraic curves of genus at most one. It develops a resolution-of-singularities framework to define well-posed initial-value spaces, introduces an explicit Lax-pair-based construction of integrals of motion $I_r$ when $s$ is a root of unity, and demonstrates that all orbits lie on fibres of $I_r$ that are generically elliptic. Extensive computational data (via Magma) for $2\le q\le 499$ support a genus-one conjecture for the fibres and reveal structured orbit-length distributions, partitioned into bins determined by the order $r$ of $s$. The results bridge arithmetic dynamics over finite fields with the geometry of elliptic curves and suggest deeper connections to zeta functions and the global dynamics of discrete Painlevé equations.

Abstract

We consider the orbits of a discrete Painlevé 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 Painlevé equations over $\mathbb{C}$, whose generic solutions are believed to be more transcendental than elliptic functions.

Arithmetic dynamics of a discrete Painlevé equation

TL;DR

The work investigates the dynamics of the discrete Painlevé equation over finite fields, showing that orbit lengths satisfy a Hasse-type bound and that orbits reside on explicitly described algebraic curves of genus at most one. It develops a resolution-of-singularities framework to define well-posed initial-value spaces, introduces an explicit Lax-pair-based construction of integrals of motion when is a root of unity, and demonstrates that all orbits lie on fibres of that are generically elliptic. Extensive computational data (via Magma) for support a genus-one conjecture for the fibres and reveal structured orbit-length distributions, partitioned into bins determined by the order of . The results bridge arithmetic dynamics over finite fields with the geometry of elliptic curves and suggest deeper connections to zeta functions and the global dynamics of discrete Painlevé equations.

Abstract

We consider the orbits of a discrete Painlevé 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 Painlevé equations over , whose generic solutions are believed to be more transcendental than elliptic functions.

Paper Structure

This paper contains 16 sections, 1 theorem, 65 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Main results
  3. Example
  4. Code and data
  5. Outline of the paper
  6. Acknowledgments
  7. Counting points on orbits
  8. Resolution of Singularities
  9. Evolution and orbits
  10. Examples
  11. Data on orbit lengths
  12. An orbit length distribution
  13. Integrals of motion
  14. Construction of integrals of motion
  15. Genera of fibres
  16. ...and 1 more sections

Key Result

Theorem 3.1

Let $r\geq 1$ be any positive integer, $R=\mathbb{Z}[s]/(\Phi_r(s))$ denote the $r$th cyclotomic integers and $Q$ its field of fractions. Then the Laurent polynomial $I_r\in R[x^{\pm 1},y^{\pm 1},t]$, defined by the trace where with defines an integral of motion of $q\textrm{P}_{\textrm{I}}$, i.e. $\overline{I}_r=I_r$ in $Q(x,y,t)$, that is of bidegree $(2r,2r)$ in $(x,y)$.

Figures (4)

  • Figure 1.1: In each subfigure, for the value of $r$ indicated in the corresponding caption, Hasse's upper bound $q+2\sqrt{q}+1$ is given by the top most solid red curve. Upper and lower bounds for the bins $B_m^{(q)}$, $1\leq m\leq 4$, are displayed alternately in solid and dashed red lines. The different numbers of points on orbits divided by $r$ are shown in blue, for all orbits and all parameter values, for any given prime power $2\leq q\leq 500$ such that $r\,\bigm |\,(q-1)$.
  • Figure 1.2: Figures in \ref{['fig:conjI']} zoomed in near the origin.
  • Figure 2.1: Schematic representation of the mapping Equation \ref{['eq:qp1']} in $X_{t,s}$.
  • Figure 2.2: In these plots, for $q=p=499$, $t=1$ and $s=140$, which has multiplicative order $r=6$ in $\mathbb{F}_q$, in blue the absolute frequency $f$ of orbit lengths over $r$ for all choices of initial values as well as the six bins \ref{['eq:bins499']} demarcated alternately by solid and dashed red grid lines. In particular, taking random initial values uniformly, the plots show the unnormalised distribution of the length of the corresponding orbit over $r$. In the top-left plot the full distribution is shown, whereas the other plots show the distribution closer to the origin at different scales.

Theorems & Definitions (14)

  • Definition 1.1
  • Conjecture 1.2
  • Remark 1.3: Autonomous case
  • Definition 2.1
  • Definition 2.2
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • ...and 4 more