Computing ideals of points
\textit{H. M. Möller} and \textit{B. Buchberger} [Computer algebra, EUROCAM '82, Conf. Marseille/France 1982, Lect. Notes Comput. Sci. 144, 24-31 (1982; Zbl 0549.68026)] gave an algorithm to compute generators of the defining ideal of a finite set of points in affine space. While this algorithm is very efficient over finite fields, one may run into the problem of very large coefficients when performing this algorithm over the rationals. The first main result in the article under review is a modular version of the Buchberger-Möller algorithm to deal with this difficulty. Next, the authors consider the problem of determining generators for the homogeneous ideal of a finite set of points in projective space. This has been considered previously by \textit{M. G. Marinari, H. M. Möller}, and \textit{T. Mora} [Appl. Algebra Eng. Commun. Comput. 4, No. 2, 103-145 (1993; Zbl 0785.13009)], and by \textit{F. Cioffi} [Ric. Mat. 48, No. 1, 55-63 (1999; Zbl 0982.13015)]. For this problem, a stopping criterion is a difficulty, since the ideal is now one-dimensional instead of zero-dimensional. The authors show that a lemma (lemma 12.1) in the cited article by Marinari et al. is incorrect, and then go on to use properties of the Hilbert function to derive a stopping criterion. Finally, the authors consider the complexity of their algorithms and give examples of their algorithms implemented in CoCoA.
scientific article; zbMATH DE number 1538748
scientific article