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.

翻译：我们认为,在多球环中,美元[x_1,\dots,$x_m]美元是一个单一的理想美元。 通常通过将投影空间的方程式$(mathbb{F})加到$I(mathbb{F})和计算基数来做到这一点。 我们用同质最大理想的饱和度来提供一种替代和更有效的方法。