Detecting Algebraic Curves in Bad Position

Abstract

Two affine algebraic curves over the complex numbers are in bad position if they have at least two distinct common points on some vertical line, i.e. for some α∈C there are distinctβ1 , β2∈C such that (α,β1 ) and (α, β2) are common points of the curves. We describe a method for detecting this situation and use it to develop an algorithm for finding the common points of two algebraic curves along with their multiplicities. We also give a worst case runtime analysis for the two algorithms and include a brief comparison of the second one with an approach using Gröbner bases.