Studies on the discrete integrable equations over finite fields

TL;DR

This paper investigates discrete integrable equations over finite fields, introduces methods to define them properly, and explores their properties and solutions, extending existing theories and applying p-adic techniques.

Contribution

It presents two novel methods for defining discrete integrable equations over finite fields, including an extension of Sakai's theory and a generalized good reduction concept.

Findings

Successfully defined discrete Painlevé equations over finite fields

Established the applicability of the extended Sakai theory for initial conditions

Introduced the concept of almost good reduction as a p-adic analog of singularity confinement

Abstract

Discrete dynamical systems over finite fields are investigated and their integrability is discussed. In particular, the discrete Painlev\'{e} equations and the discrete KdV equation are defined over finite fields and their special solutions are obtained. Their investigation over the finite fields has not been done thoroughly, partly because of the indeterminacies that appear in defining the equations. In this paper we introduce two methods to well-define the equations over the finite fields and apply the methods to several classes of discrete integrable equations. One method is to extend the space of initial conditions through blowing-up at the singular points. In case of discrete Painlev\'{e} equations, we prove that an finite field analog of the Sakai theory can be applied to construct the space of initial conditions. The other method is to define the equations over the field of $p$-adic numbers and then reduce them to the finite fields. The mapping whose time evolutions commute with the reduction is said to have a `good reduction'. We generalize good reduction in order to be applied to integrable mappings. The generalized good reduction is formulated as `almost good reduction' and is proved to be a $p$-adic analog of the singularity confinement test. (*Note that this paper is intended for the author's thesis and it draws from several of our already published papers. Also note that several minor modifications have been made from the original thesis.)