Saturation and vanishing ideals

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