μ-Bases and singularities of rational planar curves

We provide a technique to detect the singularities of rational planar curves and to compute the correct order of each singularity including the infinitely near singularities without resorting to blow ups. Our approach employs the given parametrization of the curve and uses a μ-basis for the parametrization to construct two planar algebraic curves whose intersection points correspond to the parameters of the singularities including infinitely near singularities with proper multiplicity. This approach extends Abhyankar's method of t-resultants from planar polynomial curves to rational planar curves. We also derive the classical result that for a rational planar curve of degree n the sum of all the singularities with proper multiplicity is (n−1)(n−2)/2. Examples are provided to flesh out our results.