Computing monodromy groups defined by plane algebraic curves

Abstract

We present a symbolic-numeric method to compute the monodromy group of a plane algebraic curve viewed as a ramified covering space of the complex plane. Following the definition, our algorithm is based on analytic continuation of algebraic functions above paths in the complex plane. Our contribution is three-fold : first of all, we show how to use a minimum spanning tree to minimize the length of paths ; then, we propose a strategy that gives a good compromise between the number of steps and the truncation orders of Puiseux expansions, obtaining for the first time a complexity result about the number of steps; finally, we present an efficient numerical-modular algorithm to compute Puiseux expansions above critical points,which is a non trivial task.