Computation of the singularities of parametric plane curves

Given an algebraic plane curve C defined by a rational parametrization P(t), we present formulae for the computation of the degree of C, and the multiplicity of a point. Using the results presented in [Sendra, J.R., Winkler, F., 2001. Tracing index of rational curve parametrizations. Computer Aided Geometric Design 18 (8), 771–795], the formulae simply involve the computation of the degree of a rational function directly determined from P(t). Furthermore, we provide a method for computing the singularities of C and analyzing the non-ordinary ones without knowing its defining polynomial. This approach generalizes the results in [Abhyankar, S., 1990. Algebraic geometry for scientists and engineers. In: Mathematical Surveys and Monographs, vol. 35. American Mathematical Society; van den Essen, A., Yu, J.-T., 1997. The D-resultants, singularities and the degree of unfaithfulness. Proceedings of the American Mathematical Society 25, 689–695; Gutierrez, J., Rubio, R., Yu, J.-T., 2002. D-Resultant for rational functions. Proceedings of the American Mathematical Society 130 (8), 2237–2246] and [Park, H., 2002. Effective computation of singularities of parametric affine curves. Journal of Pure and Applied Algebra 173, 49–58].