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IdealAndSeparatorsOfPoints
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ideal and separators for affine points
IdealAndSeparatorsOfPoints(Points: LIST): RECORD
where Points is a list of lists of coefficients representing a set of
distinct points in affine space.
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***** NOT YET IMPLEMENTED *****
This function computes the results of
IdealOfPoints
and
SeparatorsOfPoints
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 IdealOfPoints
separators -- the result of SeparatorsOfPoints
Thus, if the result is stored in a variable with identifier X, then:
X.points will be the input list of points;
X.ideal will be the ideal of the set of points, with generators
forming the reduced Groebner basis for the ideal;
X.separators will be a list of polynomials whose i-th element will
take the value 1 on the i-th point and 0 on the others.
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;
* 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 := IdealAndSeparatorsOfPoints(Pts);
Foreach Element In gens(X.ideal) Do
PrintLn Element;
EndForeach;
For ideals and separators of points in projective space, see
IdealAndSeparatorsOfProjectivePoints
.
Use R ::= QQ[x,y];
Points := [[1, 2], [3, 4], [5, 6]];
X := IdealAndSeparatorsOfPoints(Points);
X.points;
[[1, 2], [3, 4], [5, 6]]
-------------------------------
X.ideal;
ideal(x - y + 1, y^3 - 12y^2 + 44y - 48)
-------------------------------
X.separators;
[1/8y^2 - 5/4y + 3, -1/4y^2 + 2y - 3, 1/8y^2 - 3/4y + 1]
-------------------------------
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