Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 9 (2013), 056, 8 pages      arXiv:1209.1715      http://dx.doi.org/10.3842/SIGMA.2013.056

Integrability of Discrete Equations Modulo a Prime

Masataka Kanki
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Tokyo 153-8914, Japan

Received April 24, 2013, in final form September 05, 2013; Published online September 08, 2013

Abstract
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 finite 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.

Key words: integrability test; good reduction; discrete Painlevé equation; finite field.

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