CoCoA:SeparatorsOfPoints
From ApCoCoAWiki
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.
IdealAndSeparatorsOfProjectivePoints
<type>list</type> <type>points</type>
