1209.0223 (Masataka Kanki et al.)

The space of initial conditions and the property of an almost good
reduction in discrete Painleve II equations over finite fields    [PDF]

Masataka Kanki, Jun Mada, Tetsuji Tokihiro

We investigate the discrete Painleve equations (dPII and qPII) over finite fields. We first show that they are well defined by extending the domain according to the theory of the space of initial conditions. Then we treat them over local fields (in particular the field of p-adic numbers) and observe that they have a property that is similar to the good reduction of dynamical systems over finite fields. This property is called an 'almost good reduction' (arXiv: 1206.4456). 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.

View original: http://arxiv.org/abs/1209.0223