  We develop a theory of linear Mahler systems in several variables from the
perspective of transcendence and algebraic independence, which also includes
the possibility of dealing with several systems associated with sufficiently
independent matrix transformations. Our main results go far beyond the existing
literature, also surpassing those of two unpublished preprints the authors made
available on the arXiv in 2018. The main new feature is that they apply now
without any restriction on the matrices defining the corresponding Mahler
systems. As a consequence, we settle several problems concerning expansions of
numbers in multiplicatively independent bases. For instance, we prove that no
irrational real number can be automatic in two multiplicatively independent
integer bases, and we give a new proof and a broad algebraic generalization of
Cobham's theorem in automata theory. We also provide a new proof and a
multivariate generalization of Nishioka's theorem, a landmark result in
Mahler's method.
