The heights of iterates of the discrete Painleve equations over number fields appear to grow no
faster than polynomials while the heights of generic solutions of non-integrable discrete equations
grow exponentially. This gives rise to a simple and effective numerical test for the integrability
of discrete equations. Numerical evidence and theoretical results are presented. Connections with
other tests for integrability and Vojta’s dictionary are discussed.
This pre-print has been submitted and accepted to the journal, Journal of Physics A - Mathematical and General. The definitive version: HALBURD, R.G., 2005. Diophantine integrability. Journal of Physics A- Mathematical and General, 38(16), L263-269, is available at http://stacks.iop.org/.