Note on a question of Wilf

2021/06/14 by Michael Hellus, Hellus, Michael, Anton Rechenauer +3
Computer Science · Decision Sciences · Mathematics · #11D07 (Primary) 20M14 (Secondary) #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #Scheduling and Timetabling Solutions

paper · pdf · doi:10.48550/arxiv.2106.07246

openalex publication_date 2021/06/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let S be a numerical semigroup with Frobenius number f, genus g and embedding dimension e. % In 1978 Wilf asked the question, whether (f+1-g)/(f+1)≥\frac1e. As is well known, this holds in the cases e=2 and e=3. For e≥4, we derive from results of Zhai [5] the following (substantially weaker) lower bound (f+1-g)/(f+1)gt;((2N+1)/((2N+2)(e-2)))^e with \lfloor N\rfloor=104978 . To the best of our knowledge this is the first explicit lower bound for (f+1-g)/(f+1) in terms of the embedding dimension.

Citations

• An asymptotic result concerning a question of Wilf

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