Saturation and vanishing ideals

Abstract

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 the finite set of projective rational points \({{\mathbb {X}}}\) that it defines in the projective space \({{\mathbb {P}}}^{m-1}\). We concern ourselves with the problem of computing the vanishing ideal \(I({{\mathbb {X}}})\). This is usually done by adding the equations of the projective space \(I({{\mathbb {P}}}^{m-1})\) to I and computing the radical. We give an alternative and more efficient way using the saturation with respect to the homogeneous maximal ideal.