Minimally Generating Ideals of Rational Parametric Curves in Polynomial Time

We present an algorithm for computing a minimal set of generators for the ideal of a rational parametric projective curve in polynomial time. The method exploits the availability of polynomial algorithms for the computation of minimal generators of an ideal of points and is an alternative to the existing Gröbner bases techniques for the implicitization of curves. The termination criterion is based on the Castelnuovo–Mumford regularity of a curve. The described computation also yields the Hilbert function and, hence, the Hilbert polynomial and the Poincaré series of the curves. Moreover, it can be applied to unions of rational curves. We have compared the implementation of our algorithm with the Hilbert driven elimination algorithm included in CoCoA 3.6 and Singular 1.2, obtaining, in general, significant improvements in timings.