论文标题
计数数字半群，按frobenius编号，多重性和深度计数
Counting numerical semigroups by Frobenius number, multiplicity, and depth
论文作者
论文摘要
在1990年，Backelin表明，具有常数$ C_I $ f/2} $的FROBENIUS数字$ f $近的数值半群的数量，$ c_i \ cdot 2^{f/2} $ $ c_0 $和$ c_1 $，具体取决于$ f $的均等。在本文中，我们通过证明有$ \ lfloor {（q+1）^2/4} \ rfloor^{f/（2q-2）+O（f）} $ semigroups，带有frobenius编号$ f $ f $和深度$ q $，将此结果推广到任意深度的半群中。更一般地，对于固定的$ q \ geq 3 $，我们表明，给定$（q-1）m <f <qm $，frobenius number $ f $ and Mulutlicity $ m $的数值半群数是\ [\ left（\ left \ left \ left \ lftloor \ frac \ frac {（q+2）^2}^2}^2） \ lfloor \ frac {（q + 1）^2} {4} \ right \ rfloor^{（1-α）/2} \ right）^{m + o（m）} \]其中$α= f/m-（q-1）$。除其他事项外，这些结果暗示了Backelin的结果，增强了$ C_I $的界限，表征了相对于Frobenius数的限制多样性和属的限制分布，并解决了最近对具有固定Frobenius数字的半群数量的Singhal猜想，并解决了固定的Frobenius编号和最大嵌入尺寸。
In 1990, Backelin showed that the number of numerical semigroups with Frobenius number $f$ approaches $C_i \cdot 2^{f/2}$ for constants $C_0$ and $C_1$ depending on the parity of $f$. In this paper, we generalize this result to semigroups of arbitrary depth by showing there are $\lfloor{(q+1)^2/4}\rfloor^{f/(2q-2)+o(f)}$ semigroups with Frobenius number $f$ and depth $q$. More generally, for fixed $q \geq 3$, we show that, given $(q-1)m < f < qm$, the number of numerical semigroups with Frobenius number $f$ and multiplicity $m$ is\[\left(\left\lfloor \frac{(q+2)^2}{4} \right\rfloor^{α/2} \left \lfloor \frac{(q+1)^2}{4} \right\rfloor^{(1-α)/2}\right)^{m + o(m)}\] where $α= f/m - (q-1)$. Among other things, these results imply Backelin's result, strengthen bounds on $C_i$, characterize the limiting distribution of multiplicity and genus with respect to Frobenius number, and resolve a recent conjecture of Singhal on the number of semigroups with fixed Frobenius number and maximal embedding dimension.
