  The primary goal of this paper is to provide a general multiplicity estimate.
Our main theorem allows to reduce a proof of multiplicity lemma to the study of
ideals stable under some appropriate transformation of a polynomial ring. In
particular, this result leads to a new link between the theory of polarized
algebraic dynamical systems and transcendental number theory. On the other
hand, it allows to establish an improvement of Nesterenko's conditional result
on solutions of systems of differential equations. We also deduce, under some
condition on stable varieties, the optimal multiplicity estimate in the case of
generalized Mahler's functional equations, previously studied by Mahler,
Nishioka, Topfer and others. Further, analyzing stable ideals we prove the
unconditional optimal result in the case of linear functional systems of
generalized Mahler's type. The latter result generalizes a famous theorem of
Nishioka (1986) previously conjectured by Mahler (1969), and simultaneously it
gives a counterpart in the case of functional systems for an important
unconditional result of Nesterenko (1977) concerning linear differential
systems. In summary, we provide a new universal tool for transcendental number
theory, applicable with fields of any characteristic. It opens the way to new
results on algebraic independence, as shown in Zorin (2010).
