CoCoA:GBM

GBM

intersection of ideals for zero-dimensional schemes

Description

This function computes the intersection of ideals corresponding to

zero-dimensional schemes: GBM is for affine schemes, and <ttref>HGBM</ttref> for

projective schemes. The list L must be a list of ideals. The function <ttref>IntersectionList</ttref> should be used for computing the intersection of a collection of general ideals.

The name GBM comes from the name of the algorithm used: Generalized

Buchberger-Moeller.

Example

  Use Q[x,y,z];
  I1 := IdealOfPoints([[1,2,1], [0,1,0]]);     -- a simple affine scheme
  I2 := IdealOfPoints([[1,1,1], [2,0,1]])^2;   -- another affine scheme
  GBM([I1,I2]);                                -- intersect the ideals
Ideal(xz + yz - z^2 - x - y + 1,
 z^3 - 2z^2 + z,
 yz^2 - 2yz - z^2 + y + 2z - 1,
 y^2z - y^2 - yz + y,
 xy^2 + y^3 - 2x^2 - 5xy - 5y^2 + 2z^2 + 8x + 10y - 4z - 6,
 x^2y - y^3 + 2x^2 + 2xy + 4y^2 - 3z^2 - 8x - 8y + 6z + 5,
 x^3 + y^3 - 7x^2 - 5xy - 4y^2 + 5z^2 + 16x + 10y - 10z - 7,
 y^4 - 2y^3 - 4x^2 - 8xy - 3y^2 + 4z^2 + 16x + 16y - 8z - 12)
-------------------------------

Syntax

GBM(L:LIST):IDEAL

IdealAndSeparatorsOfPoints

IdealAndSeparatorsOfProjectivePoints

IdealOfPoints

IdealOfProjectivePoints

HGBM

   <type>groebner</type>
   <type>ideal</type>
   <type>list</type>
   <type>points</type>