Non-integrability by discrete quadratures
Résumé
We give a necessary condition for integrability by discrete quadratures of systems of difference equations: the discrete variational equations along algebraic solutions must have virtually solvable Galois groups. This necessary condition a la Morales and Ramis is used in order to prove that q-analogues of Painleve I and Painleve III equations are not integrable by discrete quadratures.