Study of toric varieties in ring theory

本文讨论了是否每条仿射单项式曲线都是集论完全交的问题，该问题是由Kronecker提出的“是否每条仿射代数曲线都是集论完全交”的著名问题派生而来的。我们成功地给出了仿射N空间中单项式曲线的定义理想是由N-2个二项式和1个到根号的多项式生成的等价条件。我们之所以考虑这个条件，是因为以前的大多数集论完全交单项式曲线的例子都满足这个条件。应用此条件，由17、19、25、27定义的单项式曲线不满足此条件。因此，它的定义理想不是由两个二项式和一个直至根号的多项式生成的。另一方面，我们可以证明它是一个集合论的完全交。事实上，它是由一个二项式和两个多项式直到根号生成的。进一步，我们可以将这个结果推广如下：如果一条单项式曲线与一个平衡半群相关联，则它是一个集论完全交。平衡半群是由A、b、c、d四个自然数定义的可加半群，其中A +d=b+c。因此，前面的例子与平衡半群有关。此外，对于与扩展平衡半群相关的单项式曲线，我们也可以证明同样的结果。

We consider a problem whether every affine monomial curve is a set-theoretic complete intersection, which is derived from a famous problem, asked by Kronecker, whether every affine algebraic curve is a set-theoretic complete intersection. We successfully give an equivalent condition in which the defining ideal of a monomial curve in affine N-space is generated by N-2 binomials and one polynomial up to radical, where N>2. The reason why we consider this condition is that most of earlier examples of set-theoretic complete intersection monomial curves satisfy this. Applying this condition, the monomial curve defined by 17, 19, 25 and 27 does not satisfy this condition. Consequently, its defining ideal is not generated by two binomials and a polynomial up to radical. On the other hand, we can prove that it is a set-theoretic complete intersection. Indeed, it is generated by one binomial and two polynomials up to radical. Further, we can extend this result as follows: If a monomial curve associated with a balanced semigroup, then it is a set-theoretic complete intersection. A balanced semigroup is an additive semigroup defined by four natural numbers, say a, b, c and d, with a+d=b+c Thus the previous example is associated with a balanced semigroup. In addition, we may prove the same result for monomial curves associated with extended balanced semigroups.