The space of initial conditions and the property of an almost good reduction in discrete Painleve II equations over finite fields

TL;DR

This paper investigates the properties of discrete Painleve II equations over finite fields, introducing the concept of 'almost good reduction' as an arithmetic analogue of singularity confinement, and explores their solutions and well-definedness.

Contribution

It extends the theory of initial conditions for discrete Painleve equations over finite fields and introduces 'almost good reduction' as a new property related to their arithmetic behavior.

Findings

Discrete Painleve II equations are well defined over finite fields using the space of initial conditions.

The concept of 'almost good reduction' is introduced as an arithmetic analogue of singularity confinement.

Special solutions over finite fields are obtained from those over characteristic zero fields.

Abstract

Discrete versions of the Painleve equations (dPII and qPII) over finite fields are studied. We first show that they are well defined by extending the domain according to the theory of the space of initial conditions, taking the dPII equation as an example. Then we define them over the field of p-adic numbers and see that they have a property that is similar to the good reduction of dynamical systems modulo a prime. This property is called 'almost good reduction'. We study the q-discrete analogue of the Painleve II equation in this paper, following the method in our previous work (arXiv: 1206.4456), in which the discrete Painleve II equation has been treated. We can consider almost good reduction as an arithmetic analogue of the singularity confinement test. We can also obtain special solutions over finite fields from those defined over fields of characteristic zero. (v2: excluded the review of arXiv: 1206.4456) (v3: several typos are corrected) (v4: final version to appear in J. Nonlin. Math. Phys.)