The minimal resultant locus

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.