IdealAndSeparatorsOfProjectivePoints

IdealAndSeparatorsOfProjectivePoints

ideal and separators for points

Syntax

IdealAndSeparatorsOfProjectivePoints(Points: LIST): RECORD

where Points is a list of lists of coefficients representing a set of
distinct points in projective space.

Description

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

This function computes the results of IdealOfProjectivePoints and SeparatorsOfProjectivePoints together at a cost lower than making the two separate calls. The result is a record with three fields:

    points     -- the points given as argument
    ideal      -- the result of IdealOfProjectivePoints
    separators -- the result of SeparatorsOfProjectivePoints

Thus, if the result is stored in a variable with identifier X, then: X.ideal will be the ideal of the set of points, with generators forming a reduced Groebner basis for the ideal; X.separators will be a list of homogeneous polynomials whose i-th element will be non-zero (actually 1, using the given representatives for the coordinates of the points) on the i-th point and 0 on the others.

NOTE:

* the current ring must have at least one more indeterminate than the dimension of the projective space in which the points lie, i.e, at least as many indeterminates as the length of an element of the input, Points;

* the base field for the space in which the points lie is taken to be the coefficient ring, which should be a field;

* in the polynomials returned, the first coordinate in the space is taken to correspond to the first indeterminate, the second to the second, and so on;

* if the number of points is large, say 100 or more, the returned value can be very large. To avoid possible problems when printing such values as a single item we recommend printing out the elements one at a time as in this example:

     X := IdealAndSeparatorsOfProjectivePoints(Pts);
     Foreach Element In gens(X.ideal) Do
       PrintLn Element;
     EndForeach;

For ideals and separators of points in affine space, see IdealAndSeparatorsOfPoints .

Example

  Use R ::= QQ[x,y,z];
  Points := [[0,0,1],[1/2,1,1],[0,1,0]];
  X := IdealAndSeparatorsOfProjectivePoints(Points);
  X.points;
[[0, 0, 1], [1, 1, 1], [0, 1, 0]]
-------------------------------
  X.ideal;
ideal(xz - 1/2yz, xy - 1/2yz, x^2 - 1/4yz, y^2z - yz^2)
-------------------------------
  X.separators;
[-2x + z, x, -2x + y]
-------------------------------

  Use R ::= QQ[t,x,y,z];
  Pts := GenericPoints(20);  -- 20 random points in projective 3-space
  X := IdealAndSeparatorsOfProjectivePoints(Pts);
  Len(Gens(X.Ideal));  -- number of generators in the ideal
17
-------------------------------
  HilbertFn(R/X.Ideal);
H(0) = 1
H(1) = 4
H(2) = 10
H(t) = 20   for t >= 3
-------------------------------
  F := X.Separators[3];
  [Eval(F, P)| P In Pts];
[0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
-------------------------------
  Res(R/X.Ideal);  -- the resolution of the ideal
0 --> R^10(-6) --> R^24(-5) --> R^15(-4) --> R
-------------------------------

See Also