Vanishing Ideals of Parameterized Subgroups in a toric variety

TL;DR

This paper presents an algorithm to compute the vanishing ideal of parameterized subgroups in toric varieties over finite fields, aiding the study of algebraic geometric codes with explicit lattice descriptions.

Contribution

It introduces an elimination-based algorithm for generators of vanishing ideals of parameterized subgroups and provides methods to compute associated lattices, with applications to algebraic geometric codes.

Findings

Algorithm for generators of vanishing ideals

Method to compute associated lattices

Nullstellensatz type theorem over finite fields

Abstract

Let $\K$ be a finite field and $X$ be a complete simplicial toric variety over $\K$. 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 \verb|Macaulay2| codes for our algorithms.