Vanishing Ideals over Finite Fields

Abstract

Let \(\mathbb{F}_{q}\) be a finite field, let \(\mathbb{X}\) be a subset of the projective space 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.