A symbolic algorithm for calculating power series expansions and dual intersection graphs of semistable models

Abstract

In this paper we develop a symbolic algorithm to calculate multivariate power series expansions of univariate polynomials over general base rings. We use this to give a complete power series algorithm to calculate the dual intersection graph of a semistable model of a curve over a non-archimedean field. We first study the problem of recovering the relative poset structure of a finite covering $X^{\prime }\to X$ of normal, relatively unibranch, Noetherian connected schemes. We show that we can reconstruct the poset structure of $X^{\prime }$ in terms of group-theoretic data over the base $X$. This group-theoretic data consists of glued double cosets, and we show how these can be interpreted in terms of glued power series approximations. We then show how our algorithms calculate these glued power series approximations, so that we can work with the branches of normalizations $X^{\prime }\to X$ without calculating integral closures. These algorithms have been implemented in OSCAR. We give a detailed study of the key steps in these algorithms for coverings of semistable models, with various examples to illustrate the non-trivial gluing phenomena. We conclude by interpreting these techniques in the context of analytic spaces, with an eye towards future applications in $p$-adic integration theory.

66 pages, 9 figures. The introduction has been rewritten and some proofs/results/definitions have been updated. The OSCAR code for the power-series algorithms in this paper can be found here: https://github.com/PaulHelminck/Newton-Puiseux