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

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 \(a < b-1\).