An in depth analysis, via resultants, of the singularities of a parametric curve (Q669015)

The knowledge of the singular locus of an algebraic curve is essential for a variety of algorithms. Given a defining polynomial $p(x,y)$ for the curve, we simply have to determine the common zeros of $p$ and its partial derivatives. The situation is more complicated if the curve is given by a rational parametrization; i.e., a pair of rational functions s.t. the curve is the Zariski closure of the image. In this paper a so-called T-function is determined by means of a resultant computation on the parametrization. By factorizing this T-function one can obtain the singularities (ordinary or non-ordinary) together with their multiplicities.