Computing zero-dimensional schemes

This paper is a natural continuation of Abbott et al. [Abbott, J., Bigatti, A., Kreuzer, M., Robbiano, L., 2000. Computing ideals of points. J. Symbolic Comput. 30, 341-356] further generalizing the Buchberger-Moller algorithm to zero-dimensional schemes in both affine and projective spaces. We also introduce a new, general way of viewing the problems which can be solved by the algorithm: an approach which looks to be readily applicable in several areas. Implementation issues are also addressed, especially for computations over Q where a trace-lifting paradigm is employed. We give a complexity analysis of the new algorithm for fat points in affine space over Q. Tables of timings show the new algorithm to be efficient in practice.

[1]  Martin Kreuzer,et al.  Computing Ideals of Points , 2000, J. Symb. Comput..

[2]  Bruno Buchberger,et al.  The Construction of Multivariate Polynomials with Preassigned Zeros , 1982, EUROCAM.

[3]  Francesca Cioffi,et al.  Computation of minimal generators of ideals of fat points , 2001, ISSAC '01.

[4]  Lorenzo Robbiano.  Zero-dimensional ideals or the inestimable value of estimable terms , 2001 .

[5]  Martin Kreuzer,et al.  Computational Commutative Algebra 1 , 2000 .

[6]  N. Bose.  Applied multidimensional systems theory , 1982 .

[7]  Maria Grazia Marinari,et al.  Gröbner bases of ideals defined by functionals with an application to ideals of projective points , 1993, Applicable Algebra in Engineering, Communication and Computing.

[8]  Jean-Charles Faugère,et al.  Efficient Computation of Zero-Dimensional Gröbner Bases by Change of Ordering , 1993, J. Symb. Comput..

[9]  Bernard Mourrain,et al.  A New Criterion for Normal Form Algorithms , 1999, AAECC.

[10]  Massimo Caboara,et al.  Families of estimable terms , 2001, ISSAC '01.

[11]  Y. N. Lakshman.  A Single Exponential Bound on the Complexity of Computing Gröbner Bases of Zero Dimensional Ideals , 1991 .

[12]  Roberto La Scala,et al.  Computing Toric Ideals , 1999, J. Symb. Comput..

[13]  N. Bose.  Multidimensional Systems Theory , 1985 .

[14]  Martin Kreuzer,et al.  On the Canonical Module of A 0-Dimensional Scheme , 1994, Canadian Journal of Mathematics.

[15]  N. Bose.  Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory , 1995 .