Vanishing ideals of parameterized subgroups in a toric variety

Abstract

Let $K$ be a finite field and $X$ be a complete simplicial toric variety over $K$ with split torus $T_X\cong (K^*)^n$. We give an algorithm relying on elimination theory for finding generators of the vanishing ideal of a subgroup $Y_Q$ parameterized by a matrix $Q$ which can be used to study algebraic geometric codes arising from $Y_Q$. We give a method to compute the lattice $L$ whose ideal $I_L$ is exactly $I(Y_Q)$ under a mild condition. As applications, we give precise descriptions for the lattices corresponding to some special subgroups. We also prove a Nullstellensatz type theorem valid over finite fields, and share Macaulay2* codes for our algorithms.