arXiv
Open Access
2021
Note on a question of Wilf
Michael Hellus
Anton Rechenauer
Rolf Waldi
Abstrak
Let $S$ be a numerical semigroup with Frobenius number $f$, genus $g$ and embedding dimension $e$. % In 1978 Wilf asked the question, whether $\frac{f+1-g}{f+1}\geq\frac1e$. As is well known, this holds in the cases $e=2$ and $e=3$. For $e\geq4$, we derive from results of Zhai [5] the following (substantially weaker) lower bound \[\frac{f+1-g}{f+1}>\left(\frac{2N+1}{(2N+2)(e-2)}\right)^e\text{ with }\lfloor N\rfloor=104978\,.\] To the best of our knowledge this is the first explicit lower bound for $\frac{f+1-g}{f+1}$ in terms of the embedding dimension.
Topik & Kata Kunci
Penulis (3)
M
Michael Hellus
A
Anton Rechenauer
R
Rolf Waldi
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