Abstract : The density theorem of Schlesinger ensures that the monodromy group of a differential system with regular singular points is Zariski-dense in its differential Galois group. We have analogs of this result for difference systems such as q-difference and Mahler systems, whose only assumption is the regular singular condition. Moreover, solutions of difference or differential systems with regular singularities have good analytical properties. For example, the solutions of differential systems which are regular singular at 0 have moderate growth at 0. We have general algorithms for recognizing regular singularities and they apply to many systems such as differential and q-difference systems. However, they do not apply to the Mahler case, systems that appear in many areas like automata theory. We will explain how to recognize regular singularities of Mahler systems. It is a joint work with Colin Faverjon.

Keywords : Mahler equations; regular singularity; algorithm