In this paper, we study a family of binomial ideals defining monomial curves in the n-dimensional affine space determined by n hypersurfaces of the form $x_i^{c_i} - x_1^{u_{i1}} \cdots x_n^{u_{1n}}$ in $\Bbbk [x_1, \ldots , x_n]$ with $u_{ii} = 0, \ i\in \{ 1, \ldots , n\}$ . We prove that the monomial curves in that family are set-theoretic complete intersections. Moreover, if the monomial curve is irreducible, we compute some invariants such as genus, type and Frobenius number of the corresponding numerical semigroup. We also describe a method to produce set-theoretic complete intersection semigroup ideals of arbitrary large height.

中文翻译：

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NORTHCOTT 类型的关键二项式理想

在本文中，我们研究了定义单项式曲线的二项式理想族n-维仿射空间由下式确定n形式的超曲面$x_i^{c_i} - x_1^{u_{i1}} \cdots x_n^{u_{1n}}$在$\Bbbk [x_1, \ldots , x_n]$和$u_{ii} = 0, \i\in \{ 1, \ldots , n\}$. 我们证明了该族中的单项式曲线是集合论完全交集。此外，如果单项式曲线是不可约的，我们计算一些不变量，如相应的数值半群的属、类型和 Frobenius 数。我们还描述了一种产生任意大高度的集合论完全交半群理想的方法。