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Dedicated to Wolfgang Schmidt on the occasion of his sixtieth birthday. Abstract. In this paper we present a rened version of the Newton polygon process to compute the Puiseux expansions of an algebraic function dened over the rational function eld. We determine an upper bound for the bit- complexity of computing the singular part of a Puiseux expansion by this algorithm, and use a recent quantitative version of Eisenstein's theorem on power series expansions of algebraic functions to show that this computational complexity is polynomial in the degrees and the logarithm of the height of the polynomial dening the algebraic function.

A polynomial-time complexity bound for the computation of the singular part of a Puiseux expansion of an algebraic function