Vanishing Ideals over Finite Fields

Let $\mathbb{F}_q$ be a finite field, let $\mathbb{X}$ be a subset of the projective space ${\mathbb P}^{s-1}$ over $\mathbb{F}_q$ parametrized by rational functions, and let $I(\mathbb{X})$ be the vanishing ideal of $\mathbb{X}$. The main result of this paper is a formula for $I(\mathbb{X})$ that will allow us to compute (i) the algebraic invariants of $I(\mathbb{X})$ and (ii) the basic parameters of the corresponding Reed–Muller-type code.