Study of toric varieties in ring theory

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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.