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HGBM

intersection of ideals for zero-dimensional schemes

Syntax
HGBM(L: LIST): IDEAL

Description
***** NOT YET IMPLEMENTED *****

This function computes the intersection of ideals corresponding to zero-dimensional schemes: GBM is for affine schemes, and HGBM for projective schemes. The list L must be a list of ideals. The function IntersectList 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. The prefix H comes from Homogeneous since ideals of projective schemes are necessarily homogeneous.

Example
  Use P := QQ[x[0..2]];
  I1 := IdealOfProjectivePoints(P, [[1,2,1], [0,1,0]]);   -- simple projective scheme
  I2 := IdealOfProjectivePoints(P, [[1,1,1], [2,0,1]])^2; -- another projective scheme
  HGBM([I1, I2]);                                      -- intersect the ideals
ideal(x[0]^3 - x[0]x[1]^2 - 5x[0]^2x[2] + x[1]^2x[2] + 8x[0]x[2]^2 - 4x[2]^3,
 x[0]^2x[1] + x[0]x[1]^2 - 3x[0]x[1]x[2] - x[1]^2x[2] + 2x[1]x[2]^2,
 x[0]x[1]^3 - 2x[0]^2x[2]^2 - 5x[0]x[1]x[2]^2 - 4x[1]^2x[2]^2 +
8x[0]x[2]^3 + 10x[1]x[2]^3 - 8x[2]^4,
 x[0]x[1]^2x[2] + x[1]^3x[2] - 2x[0]^2x[2]^2 - 5x[0]x[1]x[2]^2
- 5x[1]^2x[2]^2 + 8x[0]x[2]^3 + 10x[1]x[2]^3 - 8x[2]^4,
 x[1]^4x[2] - 2x[1]^3x[2]^2 - 4x[0]^2x[2]^3 - 8x[0]x[1]x[2]^3
- 3x[1]^2x[2]^3 + 16x[0]x[2]^4 + 16x[1]x[2]^4 - 16x[2]^5)
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See Also