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.
