Journal of Nonlinear Mathematical Physics

Volume 20, Issue Supplement 1, November 2013, Pages 101 - 109

The space of initial conditions and the property of an almost good reduction in discrete Painlevé II equations over finite fields

Authors
Masataka Kanki
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Tokyo 153-8914, Japan, kanki@ms.u-tokyo.ac.jp
Jun Mada
College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba 275-8576, Japan, mada.jun@nihon-u.ac.jp
Tetsuji Tokihiro
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Tokyo 153-8914, Japan, toki@ms.u-tokyo.ac.jp
Received 31 May 2012, Accepted 27 May 2013, Available Online 6 January 2021.
DOI
https://doi.org/10.1080/14029251.2013.862437How to use a DOI?
Keywords
discrete Painlevé equation, finite field, good reduction, space of initial condition
Abstract

We investigate the discrete Painlevé 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 and observe that they have a property that is similar to the 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.

Copyright
© 2013 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Download article (PDF)

Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
20 - Supplement 1
Pages
101 - 109
Publication Date
2021/01
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.1080/14029251.2013.862437How to use a DOI?
Copyright
© 2013 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Masataka Kanki
AU  - Jun Mada
AU  - Tetsuji Tokihiro
PY  - 2021
DA  - 2021/01
TI  - The space of initial conditions and the property of an almost good reduction in discrete Painlevé II equations over finite fields
JO  - Journal of Nonlinear Mathematical Physics
SP  - 101
EP  - 109
VL  - 20
IS  - Supplement 1
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2013.862437
DO  - https://doi.org/10.1080/14029251.2013.862437
ID  - Kanki2021
ER  -