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 distinct β_1, β_2 ∈ C such that (α,β_1) and (α, β_2) 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 brief Comparison of the second one with an approach using Grobner bases.
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