The Minimal Resultant Locus

TL;DR

This paper presents an algorithm to determine potential good reduction of rational functions over nonarchimedean fields using Berkovich geometry, characterizing the minimal resultant locus and providing bounds on field extensions.

Contribution

It introduces a geometric reformulation for detecting potential good reduction and describes the minimal resultant locus within the Berkovich projective line, with complexity bounds for rational functions over number fields.

Findings

The minimal resultant locus is either a point or a segment in the Berkovich line.

The locus is contained in the tree spanned by fixed points and poles of f(z).

The algorithm runs in probabilistic polynomial time over the rationals.

Abstract

Let K be a complete, algebraically closed nonarchimedean valued field, and let f(z) in K(z) be a rational function of degree d at least 2. We give an algorithm to determine whether f(z) has potential good reduction over K, based on a geometric reformulation of the problem using the Berkovich Projective Line. We show the minimal resultant is is either achieved at a single point in the Berkovich line, or on a segment, and that minimal resultant locus is contained in the tree in spanned by the fixed points and the poles of f(z). When f(z) is defined over the rationals, the algorithm runs in probabilistic polynomial time. If f(z) has potential good reduction, and is defined over a subfield H of K, we show there is an extension L/H in K with degree at most (d + 1)^2 such that f(z) achieves good reduction over L.