Minimally Generating Ideals of Rational Parametric Curves in Polynomial Time

The problem of computing a minimal set of generators for the ideal of a rational parametric projective curve has been tackled by elimination theory and can be solved by the computation of Gröbner bases (see, for example Kalkbrener, 1991; Licciardi and Mora, 1994; Gao and Chou, 1992; Fix et al., 1996, and references therein). These methods have been implemented in CoCoA 3.6 (Capani et al., 1995) and Singular 1.2 (Greuel et al., 1998) and have exploited the Hilbert driven algorithm (Traverso, 1996). In this paper, developing ideas of Ramella (1994), Cioffi (1996, 1999) and Orecchia (1998) we present an alternative method based on the computation of the generators for the ideal of a suitably chosen finite set of points on the curve. This has been made possible due to the availability of algorithms that construct a minimal set of generators of ideals of projective points in polynomial time (Ramella, 1990, for points in generic position; Marinari et al., 1993; Cioffi, 1999). Let C be the union of h rational curves represented parametrically over a field K which contains at least d(d ·m+1) distinct elements, where d is the degree of C and m is an upper bound for the Castelnuovo–Mumford regularity of I(C). The algorithm presented in this paper actually determines a minimal set of generators and computes the Hilbert function (then the Hilbert polynomial and the Poincaré series) of the ideal I(C). The procedure described is performed in a time polynomial in the degree of the curve and in the minimal dimension of a linear variety containing the curve. The algorithm becomes efficient by using a good termination criterion for the minimal generators based on finding a suitable bound for the Castelnuovo–Mumford regularity of the curves. For smooth curves this is given in Orecchia (1998). For singular curves a good bound for the regularity is developed in this paper and is based on the