IdealOfPoints

IdealOfPoints -- ideal of a set of affine points

Syntax

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

Description

This function computes the reduced Groebner basis for the ideal of all 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 as many indeterminates as the dimension of the 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; For ideals of points in projective space, see IdealOfProjectivePoints .

Example

/**/  use P ::= QQ[x,y];
/**/  Points := mat([[1, 2], [3, 4], [5, 6]]);
/**/  I := IdealOfPoints(P, Points);
/**/  I;
ideal(x -y +1, y^3 -12*y^2 +44*y -48)

/**/  K := NewFractionField(NewPolyRing(QQ, "a"));
/**/  use K;
/**/  Points := mat([[1,2,0], [3,4,a], [5,1,6]]);
/**/  use P ::= K[x,y,z], Lex;
/**/  I := IdealOfPoints(P, Points);
/**/  indent(I);
ideal(
  z^3 +(-a -6)*z^2 +(6*a)*z,
  y +((-a -12)/(6*a^2 -36*a))*z^2 +((a^2 +72)/(6*a^2 -36*a))*z -2,
  x +((2*a -6)/(3*a^2 -18*a))*z^2 +((-2*a^2 +36)/(3*a^2 -18*a))*z -1
)

See Also