Sub-Fibonacci behavior in numerical semigroup enumeration
Sub-Fibonacci behavior in numerical semigroup enumeration
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数值半群枚举中的次斐波那契行为
DOI：
10.5070/c63261988
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
Daniel G. Zhu
中科院分区：
文献类型：
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作者：
Daniel G. Zhu
2013年，Zhai证明了给定属的大多数数值半群的深度最多为$3$，并且属$g$的数值半群的数量$n_g$渐近于$S\varphi^g$，其中$S$是某个正常数，$\varphi \approx 1.61803$是黄金比例。在本文中，我们证明了导致 $n_g$ 偏离完美指数的因素的指数上限和下限，包括深度至少为 $4$ 的半群的数量。在其他应用中，这些结果暗示了 $n_g$ 上已知的最尖锐的渐近界限，并揭示了 Bras-Amor\'os (2008) 的猜想，即 $n_g \geq n_{g-1} + n_{g-2}$。我们的主要工具是使用 Kunz 坐标，由 Kunz (1987) 引入，以及由 Zhu (2011) 有界加权图同态得出的结果。
In 2013, Zhai proved that most numerical semigroups of a given genus have depth at most $3$ and that the number $n_g$ of numerical semigroups of a genus $g$ is asymptotic to $S\varphi^g$, where $S$ is some positive constant and $\varphi \approx 1.61803$ is the golden ratio. In this paper, we prove exponential upper and lower bounds on the factors that cause $n_g$ to deviate from a perfect exponential, including the number of semigroups with depth at least $4$. Among other applications, these results imply the sharpest known asymptotic bounds on $n_g$ and shed light on a conjecture by Bras-Amor\'os (2008) that $n_g \geq n_{g-1} + n_{g-2}$. Our main tools are the use of Kunz coordinates, introduced by Kunz (1987), and a result by Zhao (2011) bounding weighted graph homomorphisms.