Integrability of Discrete Equations Modulo a Prime

TL;DR

This paper demonstrates that the 'almost good reduction' criterion effectively characterizes integrability in discrete equations over finite fields, including Painlevé analogues and even some non-integrable systems.

Contribution

It extends the AGR criterion to discrete Painlevé equations and non-integrable systems, establishing its role similar to singularity confinement in finite field maps.

Findings

q-discrete Painlevé III and IV have AGR

Hietarinta-Viallet equation has AGR despite non-integrability

AGR can distinguish integrable from non-integrable discrete systems

Abstract

We apply the 'almost good reduction' (AGR) criterion, which has been introduced in our previous (arXiv:1206.4456 and arXiv:1209.0223), to several classes of discrete integrable equations. We verify our conjecture that AGR plays the same role for maps of the plane define over simple finite fields as the notion of the singularity confinement does. We first prove that q-discrete analogues of the Painlev\'e III and IV equations have AGR. We next prove that the Hietarinta-Viallet equation, a non-integrable chaotic system also has AGR.