Abstract
Let $S \subseteq \mathbb{N}$ be a numerical semigroup with multiplicity $m = \min(S \setminus \{0\})$ and conductor $c=\max(\mathbb{N} \setminus S)+1$. Let $P$ be the set of primitive elements of $S$, and let $L$ be the set of elements of $S$ which are smaller than $c$. A longstanding open question by Wilf in 1978 asks whether the inequality $|P||L| \ge c$ always holds. Among many partial results, Wilf's conjecture has been shown to hold in case $|P| \ge m/2$ by Sammartano in 2012. Using graph theory in an essential way, we extend the verification of Wilf's conjecture to the case $|P| \ge m/3$. This case covers more than $99.999\%$ of numerical semigroups of genus $g \le 45$.
Highlights
Denote N = {0, 1, 2, 3, . . . } and N+ = N \ {0} = {1, 2, 3, . . . }
For a numerical semigroup S, its gaps are the elements of N\S, its genus is g = |N\S|, its multiplicity is m = min S∗ where S∗ = S \ {0}, its Frobenius number is f = max Z \ S and its conductor is c = f + 1
Let S be a numerical semigroup with multiplicity m and minimal generating set P
Summary
Let S be a numerical semigroup with multiplicity m and minimal generating set P. As later noted by Manuel Delgado, who attended the Umea conference, an overwhelming majority of numerical semigroups satisfies the condition of Theorem 1. Delgado discovered that the condition of Theorem 1 is well suited to efficiently trim the tree of numerical semigroups while probing certain open problems concerning them [6]. This will lead to significant advances on the verification of Wilf’s conjecture by computer.
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