Effective computation of singularities of parametric affine curves

Abstract

Let k be a field of characteristic zero and f( t), g( t) be polynomials in k[ t]. For a plane curve parameterized by x= f( t), y= g( t), Abhyankar developed the notion of Taylor resultant (Mathematical Surveys and Monographs, Vol. 35, American Mathematical Society, Providence, RI, 1990) which enables one to find its singularities without knowing its defining polynomial. This concept was generalized as D-resultant by Yu and Van den Essen (Proc. Amer. Math. Soc. 125(3) (1997) 689), which works over an arbitrary field. In this paper, we extend this to a curve in affine n-space parameterized by x 1= f 1( t),…, x n = f n ( t) over an arbitrary ground field k, where f 1,…, f n ∈ k[ t]. This approach compares to the usual approach of computing the ideal of the curve first. It provides an efficient algorithm of computing the singularities of such parametric curves using Gröbner bases. Computational examples worked out by symbolic computation packages are included.