Puiseux power series solutions for systems of equations

We give an algorithm to compute term by term multivariate Puiseux series expansions of series arising as local parametrizations of zeroes of systems of algebraic equations at singular points. The algorithm is an extension of Newton's method for plane algebraic curves replacing the Newton polygon by the tropical variety of the ideal generated by the system. As a corollary we deduce a property of tropical varieties of quasi-ordinary singularities.
Comment: 19 pages. To appear in International Journal of Mathematics. Major changes: Several sections added explaining the geometrical meaning of the series solutions and a corollary about quasi-ordinary singularities