论文标题
关于广义数字半群的数量
On the number of generalized numerical semigroups
论文作者
论文摘要
令$ \ mathsf {r} _k $为$ x^k - （x+1）^{k -1} = 0 $的唯一正词。我们证明了$ n_ {g，d} $的最著名界限，$ d $ d $ dimensional gentrymized数字半群， \ [n_ {g，d}> c_d^{g^{（d-1）/d}} \ Mathsf {r} _ {2^d}^g \]对于一些常数$ C_D> 0 $，可以进行阐述。为此，我们将多重性和深度的概念扩展到广义的数值半群，并显示我们的下限对于深度2的半群而言。我们还通过引入分区标记在特殊类别的半群中显示了其他类别的范围，这将kunz单词的记录扩展到了一般环境。
Let $\mathsf{r}_k$ be the unique positive root of $x^k - (x+1)^{k-1} = 0$. We prove the best known bounds on the number $n_{g,d}$ of $d$-dimensional generalized numerical semigroups, in particular that \[n_{g,d} > C_d^{g^{(d-1)/d}} \mathsf{r}_{2^d}^g\] for some constant $C_d > 0$, which can be made explicit. To do this, we extend the notion of multiplicity and depth to generalized numerical semigroups and show our lower bound is sharp for semigroups of depth 2. We also show other bounds on special classes of semigroups by introducing partition labelings, which extend the notion of Kunz words to the general setting.
