  In this paper we use the connections between tropical algebraic geometry and
rigid analytic geometry in order to prove two main results. We use tropical
methods to prove a theorem about the Newton polygon for convergent power series
in several variables: if f_1,...,f_n are n convergent power series in n
variables with coefficients in a non-Archimedean field K, we give a formula for
the valuations and multiplicities of the common zeros of f_1,...,f_n. We use
rigid-analytic methods to show that stable complete intersections of tropical
hypersurfaces compute algebraic multiplicities even when the intersection is
not tropically proper. These results are naturally formulated and proved using
the theory of tropicalizations of rigid-analytic spaces, as introduced by
Einsiedler-Kapranov-Lind [EKL06] and Gubler [Gub07b]. We have written this
paper to be as readable as possible both to tropical and arithmetic geometers.
