IdealOfProjectivePoints

IdealOfProjectivePoints -- ideal of a set of projective points

Syntax

IdealOfPoints(P: RING, Points: MAT): IDEAL
where Points is a MAT of coefficients whose rows represent a set of
distinct points in projective space.

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 NumCols(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:

     I := IdealOfProjectivePoints(Pts);
     foreach g in gens(I) do
       println g;
     endforeach;

For ideals of points in affine space, see IdealOfPoints .

Example

/**/  use P ::= QQ[x,y,z];
/**/  I := IdealOfProjectivePoints(P, mat([[0,0,1],[1/2,1,1],[0,1,0]]));
/**/  I;  -- gens are the reduced Groebner basis
ideal(x*z +(-1/2)*y*z,  x*y +(-1/2)*y*z,  x^2 +(-1/4)*y*z,  y^2*z -y*z^2)

See Also