三元生成对称半群的数值实验及其推广
On Numerical Experiments with Symmetric Semigroups Generated by Three Elements and Their Generalization
摘要：
本文简单地解释了V. Arnold关于由三个元素生成的两个对称数值半群序列S（4，6 +4k，87-4k）和S（9，3 + 9 k，85- 9 k）的数值实验.我们给出了这些序列的数值半群S的推广（r_1^2，r_1r_2 + r_1^2k，r_3-r_1^2k），k \in{\mathbb Z}，r_1，r_2，r_3 \in{\mathbb Z}^+，r_1 \geq 2和\gcd（r_1，r_2）= \gcd（r_1，r_3）=1，并计算它们的泛Frobenius数Phi（r_1，r_2，r_3），其中k的取值范围很宽，且半群是对称的.我们证明了这类半群也有非对称表示。我们描述了这些半群的最小生成元集的约化，直到{r_1^2，r_3-r_1^2k}，并通过求解二次丢番图方程来求这些值。
英文摘要：
We give a simple explanation of numerical experiments of V. Arnold with two sequences of symmetric numerical semigroups, S(4,6+4k,87-4k) and S(9,3+9k,85-9k) generated by three elements. We present a generalization of these sequences by numerical semigroups S(r_1^2,r_1r_2+r_1^2k,r_3-r_1^2k), k\in{\mathbb Z}, r_1,r_2,r_3\in{\mathbb Z}^+, r_1\geq 2 and \gcd(r_1,r_2)=\gcd(r_1,r_3)=1, and calculate their universal Frobenius number Phi(r_1,r_2,r_3) for the wide range of k providing semigroups be symmetric. We show that this kind of semigroups admit also nonsymmetric representatives. We describe the reduction of the minimal generating sets of these semigroups up to {r_1^2,r_3-r_1^2k} for sporadic values of k and find these values by solving the quadratic Diophantine equation.
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