CoCoA:IdealOfProjectivePoints

IdealOfProjectivePoints

ideal of a set of projective points

Description

This function computes the reduced Groebner basis for the ideal of

all homogeneous polynomials which vanish at the given set of points.

It returns the ideal generated by that Groebner basis.

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:

<verbatim>

    I := IdealOfProjectivePoints(Pts);
    Foreach Element In Gens(I) Do
      PrintLn Element;
    EndForeach;

</verbatim> For ideals of points in affine space, see <ttref>IdealOfPoints</ttref>.

Example

  Use R ::= Q[x,y,z];
  I := IdealOfProjectivePoints([[0,0,1],[1/2,1,1],[0,1,0]]);
  I;
Ideal(xz - 1/2yz, xy - 1/2yz, x^2 - 1/4yz, y^2z - yz^2)
-------------------------------
  I.Gens;  -- the reduced Groebner basis
[xz - 1/2yz, xy - 1/2yz, x^2 - 1/4yz, y^2z - yz^2]
-------------------------------

Syntax

IdealOfProjectivePoints(Points:LIST):IDEAL

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

GBM

HGBM

GenericPoints

IdealAndSeparatorsOfPoints

IdealAndSeparatorsOfProjectivePoints

IdealOfPoints

Interpolate

QuotientBasis

SeparatorsOfPoints

SeparatorsOfProjectivePoints

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