CoCoA:SeparatorsOfPoints

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SeparatorsOfPoints

separators for affine points

Description

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

point a polynomial is determined whose value is 1 at that point and 0

at all the others. The separators yielded are reduced with respect to the reduced Groebner basis which would be found by <ttref>IdealOfPoints</ttref>.

NOTE:

* the current ring must have at least as many indeterminates as the
  dimension of the space in which the points lie;
* 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 := SeparatorsOfPoints(Pts);
    Foreach Element In S Do
      PrintLn Element;
    EndForeach;

</verbatim> For separators of points in projective space, see <ttref>SeparatorsOfProjectivePoints</ttref>.

Example

  Use R ::= Q[x,y];
  Points := [[1, 2], [3, 4], [5, 6]];
  S := SeparatorsOfPoints(Points);  -- compute the separators
  S;
[1/8y^2 - 5/4y + 3, -1/4y^2 + 2y - 3, 1/8y^2 - 3/4y + 1]
-------------------------------
  [[Eval(F, P) | P In Points] | F In S];  -- verify separators
[[1, 0, 0], [0, 1, 0], [0, 0, 1]]
-------------------------------

Syntax

SeparatorsOfPoints(Points:LIST):LIST

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

GenericPoints

IdealAndSeparatorsOfPoints

IdealAndSeparatorsOfProjectivePoints

IdealOfPoints

IdealOfProjectivePoints

Interpolate

SeparatorsOfProjectivePoints

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