A generalization of the Newton-Puiseux algorithm for semistable models

TL;DR

This paper presents an algorithm for computing the skeleton of tame coverings of curves over discretely valued fields, utilizing a new proof of the tame semistable reduction theorem and extending classical number theory results.

Contribution

It introduces a practical method to find extensions of prime ideals using power series, generalizing classical theorems and applying them to the computation of curve skeletons.

Findings

Algorithm for skeleton computation of tame coverings

Short proof of the tame simultaneous semistable reduction theorem

Generalizations of Kummer-Dedekind and Dedekind's theorems

Abstract

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