Diophantine Integrability

Abstract

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.

6 pages, 4 figures

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