Study of toric varieties in ring theory

由Kronecker提出的仿射代数曲线是否为集合论完全交的一个著名问题引出了仿射单项曲线是否为集合论完全交的问题。我们成功地给出了仿射N-空间中一条单项曲线的定义理想由N-2个二项式和一个直到根的多项式生成的一个等价条件，其中N&gt；2。我们考虑这个条件的原因是，以前集合论完全交单项曲线的大多数例子都满足这个条件。应用该条件，由17、19、25和27定义的单项曲线不满足该条件。因此，它的定义理想不是由两个二项式和一个直到根部的多项式生成的。另一方面，我们可以证明它是集合论完全交。事实上，它是由一个二项式和两个直到根部的多项式产生的。进一步地，我们可以将这一结果推广如下：如果一条单项曲线与一个平衡半群相联系，则它是集论完全交。平衡半群是由a、b、c和d四个自然数定义的加法半群，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.