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On numerical semigroups with at most 12 left elements

Daniel Marín-Aragón
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Abstract

For a numerical semigroup S ⊆ N with embedding dimension e, conductor c and left part L = S ∩ [0, c − 1], set W (S) = e|L| − c. In 1978 Wilf asked, in equivalent terms, whether W (S) ≥ 0 always holds, a question known since as Wilf's conjecture. Using a closely related lower bound W 0 (S) ≤ W (S), we show that if |L| ≤ 12 then W 0 (S) ≥ 0, thereby settling Wilf's conjecture in this case. This is best possible, since cases are known where |L| = 13 and W 0 (S) = −1. Wilf's conjecture remains open for |L| ≥ 13.
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Dates and versions

hal-02667442 , version 1 (31-05-2020)

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Shalom Eliahou, Daniel Marín-Aragón. On numerical semigroups with at most 12 left elements. 2020. ⟨hal-02667442⟩
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