Integrability of Discrete Equations Modulo a Prime

We apply the “almost good reduction” (AGR) criterion, which has been introduced in our previous works, 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 fiinite fields as the notion of the singularity confinement does. We first prove that $q$-discrete analogues of the Painlevé III and IV equations have AGR. We next prove that the Hietarinta–Viallet equation, a non-integrable chaotic system also has AGR.