Gröbner bases of ideals defined by functionals with an application to ideals of projective points

In this paper we study 0-dimensional polynomial ideals defined by a dual basis, i.e. as the set of polynomials which are in the kernel of a set of linear morphisms from the polynomial ring to the base field. For such ideals, we give polynomial complexity algorithms to compute a Gröbner basis, generalizing the Buchberger-Möller algorithm for computing a basis of an ideal vanishing at a set of points and the FGLM basis conversion algorithm.As an application to Algebraic Geometry, we show how to compute in polynomial time a minimal basis of an ideal of projective points.

[1]  Ralf Fröberg,et al.  How we proved that there are exactly 924 cyclic 7-roots , 1991, ISSAC '91.

[2]  H. M. Möller,et al.  New Constructive Methods in Classical Ideal Theory , 1986 .

[3]  B. Buchberger.  An Algorithmic Method in Polynomial Ideal Theory , 1985 .

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

[5]  Y. N. Lakshman,et al.  On the complexity of computing a Gröbner basis for the radical of a zero dimensional ideal , 1990, STOC '90.

[6]  B. Buchberger,et al.  Grobner Bases : An Algorithmic Method in Polynomial Ideal Theory , 1985 .

[7]  Rüdiger Gebauer,et al.  On an Installation of Buchberger's Algorithm , 1988, J. Symb. Comput..

[8]  Thomas Becker,et al.  The Chinese remainder problem, multivariate interpolation, and Gröbner bases , 1991, ISSAC '91.

[9]  Bruno Buchberger,et al.  A criterion for detecting unnecessary reductions in the construction of Groebner bases , 1979, EUROSAM.

[10]  Marc Giusti.  Complexity of Standard Bases in Projective Dimension Zero II , 1990, AAECC.

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

[12]  Patrizia M. Gianni,et al.  Algebraic Solution of Systems of Polynomial Equations Using Groebner Bases , 1987, AAECC.

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

[14]  Lorenzo Robbiano,et al.  Bounds for Degrees and Number of Elements in Gröbner Bases , 1990, AAECC.

[15]  Joos Heintz,et al.  Algorithmes – disons rapides – pour la décomposition d’une variété algébrique en composantes irréductibles et équidimensionnelles , 1991 .

[16]  임종인,et al.  Gröbner Bases와 응용 , 1995 .

[17]  Daniel Lazard,et al.  Gröbner-Bases, Gaussian elimination and resolution of systems of algebraic equations , 1983, EUROCAL.

[18]  Maria Grazia Marinari,et al.  Gröbner bases of ideals given by dual bases , 1991, ISSAC '91.