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arXiv · 2608.12531

An upper bound for the type of a numerical semigroup, and a reduction of Wilf's conjecture

Mohammad F. Marashdeh
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Abstract

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 Fr\"oberg, Gottlieb and H\"aggkvist settles it when $\typ\le\ee-1$. The Ap\'ery 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 $\typ\le\ee-1+\Xii(S)\le\ee-1+\Theta(S)$, where $\Theta(S)$ 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.

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Mohammad F. Marashdeh. 2026-08-12. An upper bound for the type of a numerical semigroup, and a reduction of Wilf's conjecture. https://arxiv.org/abs/2608.12531

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