CoCoA:SeparatorsOfProjectivePoints

SeparatorsOfProjectivePoints

separators for projective points

Description

This function computes separators for the points: that is, for each

point a homogeneous polynomial is determined whose value is non-zero at

that point and zero at all the others. (Actually, choosing the values listed in Points as representatives for the homogeneous coordinates of the corresponding points in projective space, the non-zero value will be 1.) The separators yielded are reduced with respect to the reduced Groebner basis which would be found by <ttref>IdealOfProjectivePoints</ttref>.

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;

* the separators are in the same order as the points (i.e. the first
  separator is the one corresponding the first point, 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>

    S := SeparatorsOfProjectivePoints(Pts);
    Foreach Element In S Do
      PrintLn Element;
    EndForeach;

</verbatim> For separators of points in affine space, see <ttref>SeparatorsOfPoints</ttref>.

Example

  Use R ::= Q[x,y,z];
  Points := [[0,0,1],[1/2,1,1],[0,1,0]];
  S := SeparatorsOfProjectivePoints(Points);
  S;
[-2x + z, 2x, -2x + y]
-------------------------------
  [[Eval(F, P) | P In Points] | F In S];   -- verify separators
[[1, 0, 0], [0, 1, 0], [0, 0, 1]]
-------------------------------

Syntax

SeparatorsOfProjectivePoints(Points:LIST):LIST

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

GenericPoints

IdealAndSeparatorsOfPoints

IdealAndSeparatorsOfProjectivePoints

IdealOfPoints

IdealOfProjectivePoints

Interpolate

SeparatorsOfPoints

   <type>list</type>
   <type>points</type>