关于一类数字半群的Frobenius问题

Frobenius Problem of a Class of Numeric Semigroups

给出Frobenius数只依赖于该数字半群的极小生成元系的计算公式是著名的Frobenius问题。目前,由一些特殊数列生成的数字半群的Frobenius问题,如Thabit数字半群等的Frobenius问题已经得到了解决。现对Gu等人研究的数字半群进行推广,通过归纳和定理证明,确定了推广后的新数字半群的嵌入维数的计算公式及新数字半群的极小生成元系,并探究了新数字半群的Apéry集等相关性质,确定了两种特殊情况下的新数字半群的Apéry集和Frobenius数的计算公式。

The Frobenius problem is to find the formula for the Frobenius number of the numerical semigroup.At present,the Frobenius problem of the numerical semigroup generated by some special sequences,such as the Frobenius problem of Thabit numerical semigroups has been solved.The numerical semigroup studied by Gu Z et al is genetralized.Through induction and theorem proving,the embedding dimension of the new numerical semigroup after extension is determined,and the minimal generators system of the new numerical semigroup is also determined.Some properties of new numerical semigroups,such as the Apéry sets,are also studied.The Apéry sets and the formulas for calculating Frobenius number of new numerical semigroups in two special cases are determined.