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Rolf Waldi

Publications and source records attributed to Rolf Waldi.

10 recordsLinked to original sources
arXiv2024

On the order of magnitude of certain integer sequences

Let $p$ be a prime number, and let $S$ be the numerical semigroup generated by the prime numbers not less than $p$. We compare the orders of magnitude of some invariants of $S$ with each other, e. g., the biggest atom $u$ of $S$ with $p$ itself: By Harald Helfgott (arXiv:1312.7748 [math.NT]), every odd integer $N$ greater than five can be written as the sum of three prime numbers. There is numerical evidence suggesting that the summands of $N$ always can be chosen between $\frac N6$ and $\frac N2$. This would imply that $u$ is less than $6p$.

math.NT↗
arXiv2021

A lower bound for the Wilf density, deduced from a result of Zhai

Let $S\neq\mathbb N$ 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 $e=3$. From Zhai's results in [5] we derive \[\frac{f+1-g}{f+1}\geq\frac2{e^2-e+2}\text{ for }e\geq4\,.\]

math.NT↗
arXiv2021

Note on a question of Wilf

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.

math.NT↗
arXiv2020

On the Frobenius number of certain numerical semigroups

Let $0<\lambda\leq1$, $\lambda\notin\left\{\frac24, \frac27, \frac2{10}, \frac2{13}, \ldots\right\}$, be a real and $p$ a prime number, with $[p,p+\lambda p]$ containing at least two primes. Denote by $f_\lambda(p)$ the largest integer which cannot be written as a sum of primes from $[p,p+\lambda p]$. Then \[f_\lambda(p)\sim\left\lfloor2+\frac2\lambda\right\rfloor\cdot p\text{, as }p\text{ goes to infinity.}\] Further a question of Wilf about the 'Money-Changing Problem' has a positive answer for all semigroups of multiplicity $p$ containing the primes from $[p,2p]$. In particular, this holds for the semigroup generated by all primes not less than $p$. The latter special case was already shown in a previous paper.

math.NT↗
arXiv2019

Numerical Semigroups generated by Primes

Let $p_1=2, p_2=3, p_3=5, \ldots$ be the consecutive prime numbers, $S_n$ the numerical semigroup generated by the primes not less than $p_n$ and $u_n$ the largest irredundant generator of $S_n$. We will show, that $\bullet$ $u_n\sim3p_n$. Similarly, for the largest integer $f_n$ not contained in $S_n$, by computational evidence we suspect that $\bullet$ $f_n$ is an odd number for $n\geq5$ and $\bullet$ $f_n\sim3p_n$; further $\bullet$ $4p_n>f_{n+1}$ for $n\geq1$. If $f_n$ is odd for large $n$, then $f_n\sim3p_n$. In case $f_n\sim3p_n$ every large even integer $x$ is the sum of two primes. If $4p_n>f_{n+1}$ for $n\geq1$, then the Goldbach conjecture holds true. Further, Wilf's question in [12] has a positive answer for the semigroups $S_n$.

math.NT↗
arXiv2018

Variants on a question of Wilf

Let $S\neq\mathbb N$ be a numerical semigroup generated by $e$ elements. In his paper (A Circle-Of-Lights Algorithm for the "Money-Changing Problem", Amer. Math. Monthly 85 (1978), 562--565), H.~S.~Wilf raised the following question: Let $\Omega$ be the number of positive integers not contained in $S$ and $c-1$ the largest such element. Is it true that the fraction $\frac\Omega c$ of omitted numbers is at most $1-\frac1e$? Let $B\subseteq\mathbb N^{e-1}$ be the complement of an artinian $\mathbb N^{e-1}$-ideal. Following a concept of A.~Zhai (An asymptotic result concerning a question of Wilf, arXiv:1111.2779v1 [math.CO]) we relate Wilf's problem to a more general question about the weight distribution on $B$ with respect to a positive weight vector. An affirmative answer is given in special cases, similar to those considered by R.~Fr\"oberg, C.~Gottlieb, R.~H\"aggkvist (On numerical semigroups, Semigroup Forum, Vol.~35, Issue 1, 1986/1987, 63--83) for Wilf's question.

math.AC↗
arXiv2018

Distributions of weights and a question of Wilf

Let $S$ be a numerical semigroup of embedding dimension $e$ and conductor $c$. The question of Wilf is, if $\#(\mathbb N\setminus S)/c\leq e-1/e$. \noindent In (An asymptotic result concerning a question of Wilf, arXiv:1111.2779v1 [math.CO], 2011, Lemma 3), Zhai has shown an analogous inequality for the distribution of weights $x\cdot\gamma$, $x\in\mathbb N^d$, w.\,r. to a positive weight vector $\gamma$: \noindent Let $B\subseteq\mathbb N^d$ be finite and the complement of an $\mathbb N^d$-ideal. Denote by $\operatorname{mean}(B\cdot\gamma)$ the average weight of $B$. Then \[\operatorname{mean}(B\cdot\gamma)/\max(B\cdot\gamma)\leq d/d+1.\] $\bullet$ For the family $\Delta_n:=\{x\in\mathbb N^d|x\cdot\gamma<n+1\}$ of such sets we are able to show, that $\operatorname{mean}(\Delta_n\cdot\gamma)/\max(\Delta_n\cdot\gamma)$ converges to $d/d+1$, as $n$ goes to infinity. $\bullet$ Applying Zhai's Lemma 3 to the Hilbert function of a positively graded Artinian algebra yields a new class of numerical semigroups satisfying Wilf's inequality.

math.AC↗
arXiv2011

Weierstraßsemigroups and nodal curves of type p,q

We study plane curves of type p,q having only nodes as singularities. Every Weierstraßsemigroup is the Weierstraßsemigroup of such a curve at its place at infinity for properly chosen p,q. We construct plane curves of type p,q with explicitly given nodes and determine their Weierstraßsemigroups. Many such semigroups are found.

math.AG↗
arXiv2011

The space of nodal curves of type p,q with given Weierstraßsemigroup

We continue the investigation of curves of type p,q started in [KKW]. We study the space of such curves and the space of nodal curves with prescribed Weierstraßsemigroup. A necessary and sufficient criterion for a numerical semigroup to be a Weierstraßsemigroup is given. We find a class of Weierstraßsemigroups which apparently has not yet been described in the literature.

math.AG↗