数值半群型的一个上界，以及Wilf猜想的一个归约
An upper bound for the type of a numerical semigroup, and a reduction of Wilf's conjecture
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中文总结 AI 辅助
该研究给出数值半群型的上界，将Wilf猜想归约为不含导子和nn的不等式，确定亏格不等式等号情形，修正相关分类并回答特定问题。
中文摘要 AI 辅助
设$S$为数值半群，其重数为$\text{mult}$，导子为$\text{cc}$，嵌入维数为$\text{ee}$，型为$\text{typ}$，亏格为$\text{gnus}$，令$\text{nn}=\text{cc}-\text{gnus}$。Wilf猜想断言$\text{ee}\times\text{nn}\text{≥}\text{cc}$；Fröberg、Gottlieb和Häggkvist的不等式$\text{gnus}\text{≤}\text{typ}\times\text{nn}$在$\text{typ}\text{≤}\text{ee}-1$时可证该猜想成立。$S$关于任意$s \text{∈} S\backslash\{0\}$的Apéry集带有偏序，其极大元是平移$s$的伪Frobenius数；当$s=\text{mult}$时，其极小元是除$\text{mult}$外的极小生成元。比较这两个极值统计量可将型界定为$\text{typ}\text{≤}\text{ee}-1+\text{Xii}(S)\text{≤}\text{ee}-1+\text{Theta}(S)$，其中$\text{Theta}(S)$衡量伪Frobenius数覆盖$S$的间隙的冗余度，$\text{Xii}(S)$是对其的精细化。结合Wilf数的精确分解，这给出亏格界$\text{gnus}\text{≤}\text{ee}-1+\text{typ}\times(\text{nn}-1)$，该界在$\text{typ}\text{≥}\text{ee}$时严格强于$\text{gnus}\text{≤}\text{typ}\times\text{nn}$，并将Wilf猜想归约为不含$\text{cc}$和$\text{nn}$的不等式。确定$\text{gnus}\text{≤}\text{typ}\times\text{nn}$的等号情形，得到Singhal的分类；在$\text{ee}\text{≥}\text{typ}+1$时回答Moscariello和Sammartano的问题；修正Kaplan关于$\text{cc}\text{≤}2\text{mult}$的等号情形分类，该分类缺失一个无限族。
英文摘要
Let $S$ be a numerical semigroup with multiplicity $\mult$, conductor $\cc$, embedding dimension $\ee$, type $\typ$ and genus $\gnus$, and let $\nn=\cc-\gnus$. Wilf's conjecture asserts that $\ee\,\nn\ge\cc$; the inequality $\gnus\le\typ\,\nn$ of Fröberg, Gottlieb and Häggkvist settles it when $\typ\le\ee-1$. The Apéry set of $S$ with respect to any $s\in S\setminus\{0\}$ carries a partial order whose maximal elements are the pseudo-Frobenius numbers translated by $s$; for $s=\mult$ its minimal elements are the minimal generators other than $\mult$. Comparing the two extremal statistics bounds the type by $\typ\le\ee-1+\Xii(S)\le\ee-1+Θ(S)$, where $Θ(S)$ measures the redundancy of the covering of the gaps of $S$ by the pseudo-Frobenius numbers and $\Xii(S)$ refines it. With an exact decomposition of the Wilf number this yields the genus bound $\gnus\le\ee-1+\typ(\nn-1)$, strictly stronger than $\gnus\le\typ\,\nn$ precisely when $\typ\ge\ee$, and reduces Wilf's conjecture to an inequality free of $\cc$ and $\nn$. We determine the equality case of $\gnus\le\typ\,\nn$, recovering a classification of Singhal; answer a question of Moscariello and Sammartano whenever $\ee\ge\typ+1$; and correct Kaplan's classification of the equality case for $\cc\le2\mult$, from which an infinite family is missing.
