Saturation and vanishing ideals

TL;DR

This paper introduces a more efficient method for computing the vanishing ideal of a finite set of projective points over a finite field by using saturation techniques instead of traditional radical computations.

Contribution

It proposes an alternative approach leveraging saturation with respect to the homogeneous maximal ideal for computing vanishing ideals more efficiently.

Findings

The saturation method reduces computational complexity.

It provides a practical algorithm for finite field projective point sets.

The approach improves upon traditional radical-based methods.

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.