Saturation and vanishing ideals

We consider an homogeneous ideal I in the polynomial ring S=K[x_1,… , x_m] over a finite field K=𝔽_q and the finite set of projective rational points 𝕏 that it defines in the projective space ℙ^m-1 . We concern ourselves with the problem of computing the vanishing ideal I(𝕏) . This is usually done by adding the equations of the projective space I(ℙ^m-1) to I and computing the radical. We give an alternative and more efficient way using the saturation with respect to the homogeneous maximal ideal.