# Computing minimal generating systems for some special toric ideals

## TL;DR

This paper discusses a method for efficiently computing minimal generating systems for certain toric ideals associated with lattice polytopes, simplifying the process to Gaussian elimination when boundary lattice points are at least four.

## Contribution

It provides a straightforward approach to determine minimal generators of toric ideals for specific projective toric surfaces, reducing the problem to linear algebra techniques.

## Key findings

- Minimal generating systems can be computed via Gaussian elimination.
- Applicable when the boundary of the lattice polytope has at least four lattice points.
- Simplifies the algebraic process for certain toric ideals.

## Abstract

Let $X_{P}$ be the projective toric surface associated to a lattice polytope $P$. If the number of lattice points lying on the boundary of $P$ is at least $4$, it is known that $X_{P}$ is embeddable into a suitable projective space as zero set of finitely many quadrics. In this case, the determination of a minimal generating system of the toric ideal defining $X_{P}$ is reduced to a simple Gaussian elimination.