Discrete Painleve II equation over finite fields

TL;DR

This paper studies the discrete Painleve II equation over finite fields, revealing an arithmetic property akin to good reduction that helps resolve indeterminacies and derive special solutions.

Contribution

It introduces an arithmetic analogue of singularity confinement for the discrete Painleve II equation over finite fields, enabling the avoidance of indeterminacies and construction of special solutions.

Findings

The equation exhibits a property similar to good reduction over finite fields.

This property helps avoid indeterminacies in the equation.

Special solutions can be obtained from those over characteristic zero fields.

Abstract

We investigate the discrete Painleve II equation over finite fields. We treat it over local fields and observe that it has a property that is similar to the good reduction over finite fields. We can use this property, which seems to be an arithmetic analogue of singularity confinement, to avoid the indeterminacy of the equations over finite fields and to obtain special solutions from those defined originally over fields of characteristic zero.