Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Minimal Systems of Binomial Generators for the Ideals of Certain Monomial Curves

Version 1 : Received: 5 November 2021 / Approved: 8 November 2021 / Online: 8 November 2021 (12:37:42 CET)

How to cite: Branco, M.B.; Colaço, I.; Ojeda, I. Minimal Systems of Binomial Generators for the Ideals of Certain Monomial Curves. Preprints 2021, 2021110135 (doi: 10.20944/preprints202111.0135.v1). Branco, M.B.; Colaço, I.; Ojeda, I. Minimal Systems of Binomial Generators for the Ideals of Certain Monomial Curves. Preprints 2021, 2021110135 (doi: 10.20944/preprints202111.0135.v1).

## 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 $a < b-1$.

## Keywords

Binomial ideal; semigroup ideal; minimal systems of generators; determinantal ideal; Gröbner basis; Indispensability

## Subject

MATHEMATICS & COMPUTER SCIENCE, Algebra & Number Theory

Views 0