Discrete Painlevé II equation over finite fields

Abstract

We investigate some of the discrete Painlevé equations (dP II , qP I and qP II ) and the discrete KdV equation over finite fields 1 . The first part concerns the discrete Painlevé equations. We review some of the ideas introduced in our previous papers [1,2] and give some detailed discussions. We first show that they are well defined by extending the domain according to the theory of the space of initial conditions. We then extend them to the field of p-adic numbers and observe that they have a property that is called an 'almost good reduction' of dynamical systems over finite fields. We can use this property, which can be interpreted as 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. In the second part we study the discrete KdV equation. We review the discussions in [3] and present a way to resolve the indeterminacy of the equation by treating it over a field of rational functions instead of the finite field itself. Explicit forms of soliton solutions and their periods over finite fields are obtained.

Some integrable maps and their Hirota bilinear forms

“…This can be turned into a seminumerical method for measuring the growth of complexity [1], with the rate of growth of the logarithmic heights of the iterates being taken as a measure of entropy [11]. Furthermore, if the map is defined over Q, then one can consider reduction modulo a prime p, in which case the appearance of a singularity at some iterate u n ∈ Q means that the p-adic norm |u n | p > 1, and the p-adic expansion of the iterates (expanding in powers of p) is analogous to the expansion in powers of ε in the usual singularity confinement test (see [23,24] for an application of this idea).…”

Integrability of Discrete Equations Modulo a Prime

“…We have also proved that the discrete and q-discrete Painlevé II equations also have almost good reduction [7,8]. From these observations we have conjectured that, for the map of the plane Φ n defined over the field of p-adic numbers, having almost good reduction on the domain…”

“…In the previous papers, we defined the generalized notion of good reduction so that it could be applied to wider class of integrable mappings. We called this notion "almost good reduction" (AGR), and proved that discrete and q-discrete Painlevé II equations have AGR [7,8]. Our conjecture was that AGR is also satisfied for other discrete Painlevé equations and that AGR is closely related to the integrability of dynamical systems over finite fields.…”

Tropical geometric interpretation of ultradiscrete singularity confinement

“…The approach that is closest to our framework is that of Kasman and Lafortune [18], yet they do not make any tropical geometric connections in their work. Algebraically, this approach is actually closer to the arithmetic integrability of Kanki et al [16].…”