A generalization of the Newton-Puiseux algorithm for semistable models

In this paper we give an algorithm that calculates a skeleton of a tame covering of curves over a complete discretely valued field. The algorithm mainly relies on the {tame simultaneous semistable reduction theorem}, for which we give a short proof based on first principles. To use this theorem in practice, we show that we can find extensions of prime ideals in normalizations using power series. We also prove that these power series can be used to find extensions of chains of prime ideals. This then allows us to reconstruct a skeleton of a tame covering. In studying the connections between power series and extensions of prime ideals, we furthermore obtain generalizations of classical theorems from number theory such as the Kummer-Dedekind theorem and Dedekind's theorem for cycles in Galois groups.