Minimal binomial systems of generators for the ideals of certain monomial curves

TL;DR

This paper investigates the structure of certain toric ideals associated with monomial curves, proving determinantal properties and conditions for uniqueness of minimal generators based on parameters a, b, and n.

Contribution

It establishes that specific toric ideals are determinantal and characterizes when these ideals have unique minimal generating systems depending on the parameters.

Findings

The toric ideal is determinantal.

Uniqueness of minimal generators occurs if and only if a < b-1 for n > 3.

Provides explicit conditions linking parameters to ideal properties.

Abstract

Let $a, b$ and $n > 1$ be three positive integers such that $a$ and $\sum_{j=0}^{n-1} b^j$ are relatively prime. In this paper, we prove that the toric ideal $I$ associated to the submonoid of $\mathbb{N}$ generated by $\{\sum_{j=0}^{n-1} b^j\} \cup \{\sum_{j=0}^{n-1} b^j + a\, \sum_{j=0}^{i-2} b^j \mid i = 2, \ldots, n\}$ is determinantal. Moreover, we prove that for $n > 3$, the ideal $I$ has a unique minimal system of generators if and only if $a < b-1$.