The difference Galois theory of Mahler equations is an active research area. The present paper aims at developing the analytic aspects of this theory. We first attach a pair of connection matrices to any regular singular Mahler equation. We then show that these connection matrices can be used to produce a Zariski-dense subgroup of the difference Galois group of any regular singular Mahler equation.
This paper is devoted to the study of the analytic properties of Mahler systems at 0. We give an effective characterisation of Mahler systems that are regular singular at 0, that is, systems which are equivalent to constant ones. Similar characterisations already exist for differential and (q-)difference systems but they do not apply in the Mahler case. This work fill in the gap by giving an algorithm which decides whether or not a Mahler system is regular singular at 0.
This paper is devoted to the study of the analytic properties of Mahler systems at
0
0
. We give an effective characterisation of Mahler systems that are regular singular at
0
0
, that is, systems which are equivalent to constant ones. Similar characterisations already exist for differential and (
q
q
-)difference systems but they do not apply in the Mahler case. This work fills in the gap by giving an algorithm which decides whether or not a Mahler system is regular singular at
0
0
. In particular, it gives an effective characterisation of Mahler systems to which an analog of Schlesinger’s density theorem applies.
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