Singularity confinement and algebraic entropy: the case of the discrete Painleve equations

We examine the validity of the results obtained with the singularity confin ement integrability criterion in the case of discrete Painleve equations. T he method used is based on the requirement of non-exponential growth of the homogeneous degree of the iterate of the mapping. We show that when we sta rt from an integrable autonomous mapping and deautonomise it using singular ity confinement the degrees of growth of the nonautonomous mapping and of t he autonomous one are identical. Thus this low-growth based approach is com patible with the integrability of the results obtained through singularity confinement. The origin of the singularity confinement property and its nec essary character for integrability are also analysed. (C) 1999 Published by Elsevier Science B.V. All rights reserved.