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Integrability of discrete systems over finite fields

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Discrete Integrable Systems

Discrete integrable systems defined over finite fields are studied. Dynamical systems over non-Archimedean fields are of great interest in the theory of arithmetic dynamical systems [1].Discrete integrable equations over the field of p -adic numbers are defined and then the evolutions are reduced to the finite field. The integrable systems are shown to have a property that resembles a ‘good reduction’ modulo a prime [2]. We observe that this generalization of the good reduction can be used to test integrability of discrete equations over finite fields. We discuss the relation of our methods to other integrability tests, in particular, the ‘singularity confinement test’. Applications of our approach to the two-dimensional lattice systems such as the discrete KdV equation are also studied.

[1] J. H. Silverman, The Arithmetic of Dynamical Systems, (2007), Springer-Verlag, New York. [2] M. Kanki, J. Mada, K. M. Tamizhmani, T. Tokihiro, Discrete Painleve II equation over finite fields, J. Phys. A: Math. Theor., 45, (2012), 342001 (8pp).

This talk is part of the Isaac Newton Institute Seminar Series series.

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