  This is the first part of a work devoted to the study of linear Mahler
systems in several variables from the perspective of transcendence and
algebraic independence. We prove two main results concerning systems that are
regular singular at the origin. Our interest in Mahler's method comes from the
possible applications of these results to old problems involving automata
theory and which concern the expansion of both natural numbers and real numbers
in integer bases. In particular, problems which involve finite automata and
base change. Such applications are studied in Part II of this work.
