An Isaac Newton Institute Workshop

Painleve Equations and Monodromy Problems: Recent Developments

Diophantine Integrability

18th September 2006

Author: Rod Halburd (Loughborough University)

Abstract

Discrete equations over the rational numbers (and more generally over number fields) will be considered. The height of a rational number a/b is max(a,b), where a and b are coprime. The height of the nth iterate of an equation appears to grow like a power of n for discrete equations broadly considered to be of Painlev\'e type, and exponentially for other equations. Methods for classifying equations according to this criterion will be described. Connections with other approaches, such as Nevanlinna theory, singularity confinement and algebraic entropy, will be discussed.