Use R ::= QQ[t,x,y,z];
Pts := GenericPoints(20);  20 random points in projective 3space
X := IdealAndSeparatorsOfProjectivePoints(Pts);
Len(Gens(X.Ideal));  number of generators in the ideal
17

Hilbert(R/X.Ideal);
H(0) = 1
H(1) = 4
H(2) = 10
H(t) = 20 for t >= 3

F := X.Separators[3];
[Eval(F, P) P In Pts];
[0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

Res(R/X.Ideal);  the resolution of the ideal
0 > R^10(6) > R^24(5) > R^15(4) > R

