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.

References in corpus (1)

• Projective nested cartesian codes

Cited by in corpus (1)

• Projective Nullstellensatz for not necessarily algebraically closed fields