Alessio Sammartano
We give an affirmative answer to Wilf's conjecture for numerical semigroups satisfying 2 ν\geq m, where νand m are respectively the embedding dimension and the multiplicity of a semigroup. The conjecture is also proved when m \leq 8 and when the semigroup is generated by a generalized arithmetic sequence.
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The aim of this paper is to propose an extension of the Wilf conjecture to semimodules over a numerical semigroup through a new approach toward the solution of the Wilf conjecture on numerical semigroups. The key point is the introduction of a new invariant that we call Wilf function of a semigroup that can be also defined for a semimodule over the semigroup. We also study the relation of the Wilf function of a semigroup and the Wilf function of certain semimodules associated to a gap of the semigroup. As a consequence, we obtain an alternative conjecture which also implies the Wilf conjecture…
Wilf Inequality is preserved under Gluing of Semigroups
本文证明了数值半群的Wilf不等式在半群粘合操作下保持不变，即若两个半群满足该不等式，其粘合也满足，并在高维情形下证明了类似结果。
In this work, we study good semigroups of $\mathbb{N}^n$ introducing the definition of length and genus for these objects. We show how to count the local good semigroup with a fixed genus. Furthermore, we study the relationships of these concepts with other ones previously defined in the case of good semigroups with two branches.
Eliahou number, Wilf function and concentration of a numerical semigroup
该研究围绕数值半群的Eliahou数、Wilf函数与浓度展开，给出相关估计与必要条件，构造新例子，提出高密度数值半群概念并证明其相关性质，为Wilf猜想提供新证据。
A numerical semigroup is a submonoid of $\mathbb N$ with finite complement in $\mathbb N$. A generalized numerical semigroup is a submonoid of $\mathbb{N}^{d}$ with finite complement in $\mathbb{N}^{d}$. In the context of numerical semigroups, Wilf's conjecture is a long standing open problem whose study has led to new mathematics and new ways of thinking about monoids. A natural extension of Wilf's conjecture, to the class of $\mathcal C$-semigroups, was proposed by García-García, Marín-Aragón, and Vigneron-Tenorio. In this paper, we propose a different generalization of Wilf's conjecture, to…
Wilf's question in numerical semigroups $S_3$ revisited and inequalities for rescaled genera
本文研究三元极小生成数值半群 $S_3$，借助合冲次数恒等式给出 Wilf 问题肯定答案的简短证明，并建立 Frobenius 数下界与重标度亏格的上下界。
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 $Ω$ be the number of positive integers not contained in $S$ and $c-1$ the largest such element. Is it true that the fraction $\fracΩc$ 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…
Given $m\in \mathbb{N},$ a numerical semigroup with multiplicity $m$ is called packed numerical semigroup if its minimal generating set is included in $\{m,m+1,\ldots, 2m-1\}.$ In this work, packed numerical semigroups are used to built the set of numerical semigroups with fixed multiplicity and embedding dimension, and to create a partition in this set. Moreover, Wilf's conjecture is checked in the tree associated to some packed numerical semigroups.
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 $x\cdotγ$, $x\in\mathbb N^d$, w.\,r. to a positive weight vector $γ$: \noindent Let $B\subseteq\mathbb N^d$ be finite and the complement of an $\mathbb N^d$-ideal. Denote by $\operatorname{mean}(B\cdotγ)$ the average weight of $B$. Then \[\operatorname{mean}(B\cdotγ)/\ma…
An upper bound for the multiplicity and Wilf's conjecture for one-dimensional Cohen-Macaulay rings
本文研究一维 Cohen-Macaulay 环的重数上界，刻画等号情形，联系数值半群环的 Wilf 猜想，并证明近似 Gorenstein 环上的相应猜想。
We give an algorithm to determine whether Wilf's conjecture holds for all numerical semigroups with a given multiplicity $m$, and use it to prove Wilf's conjecture holds whenever $m \le 18$. Our algorithm utilizes techniques from polyhedral geometry, and includes a parallelizable algorithm for enumerating the faces of any polyhedral cone up to orbits of an automorphism group. We also introduce a new method of verifying Wilf's conjecture via a combinatorially-flavored game played on the elements of a certain finite poset.
Numerical Semigroups with $a_e = 2g+1$
本文研究了最大生成元为 $2g+1$ 的数值半群，揭示其与对称半群的联系及自身对称性，并证明Wilf问题对这些半群有正解。
We define the concentration of a numerical semigroup $S$ as $\mathsf{C}(S)=\max \left\{\text{next}_S(s)-s ~|~ s\in S \backslash \{0\}\right\}$ wherein $\text{next}_S(s)=\min\left\{x \in S ~|~ s<x\right\}$. In this paper, we study the class of numerical semigroups with concentration $2$. We give algorithms to calculate the whole set of this class of semigroups with given multiplicity, genus or Frobenius number. Separately, we prove that this class of semigroups verifies the Wilf's conjecture.
In this paper, we introduce and study the numerical semigroups generated by $\{a_1, a_2, \ldots \} \subset \mathbb{N}$ such that $a_1$ is the repunit number in base $b > 1$ of length $n > 1$ and $a_i - a_{i-1} = a\, b^{i-2},$ for every $i \geq 2$, where $a$ is a positive integer relatively prime with $a_1$. These numerical semigroups generalize the repunit numerical semigroups among many others. We show that they have interesting properties such as being homogeneous and Wilf. Moreover, we solve the Frobenius problem for this family, by giving a closed formula for the Frobenius number in terms …
Supersymmetric gaps of a numerical semigroup with two generators
该研究针对双生成元数值半群提出超对称间隙与自对称间隙新概念，证明其集合可完全确定半群，并与基本间隙做了对比，丰富了数值半群的对称相关研究。
Affine semigroups of maximal projective dimension-II
该研究聚焦极大投射维数仿射半群，给出单纯半群为Buchsbaum的充要条件、几类半群的刻画，证明部分半群满足扩展Wilf猜想，还发现Betti型可不受嵌入维数有界约束，并探索满足Arf性质的子幺半群。
On Numerical Semigroups with Fixed Quotient
研究固定商下数值半群的纤维结构，分析其生成元、不变量及深度，并引入M_d(Δ)-秩及其相关不变量。
On the seeds and the great-grandchildren of a numerical semigroup
本文重新定义数值半群的种子算法，利用至多三个左元素的半群种子和任意半群的曾孙关系，结合位运算、并行化与深度优先搜索，在中等亏格范围内超越RGD算法，并证明亏格66的Wilf猜想及发现亏格67的新Eliahou半群。
In this note we characterize the affine semigroup rings K[S] over an arbitrary field K that satisfy condition R_l of Serre. Our characterization is in terms of the face lattice of the positive cone pos(S) of S. We start by reviewing some basic facts about the faces of pos(S) and consequences for the monomial primes of K[S]. After proving our characterization we turn our attention to the Rees algebras of a special class of monomial ideals in a polynomial ring over a field. In this special case, some of the characterizing criteria are always satisfied. We give examples of nonnormal affine semigr…
Tangent Cones of Concatenated Numerical Semigroups
本文研究等差数列拼接数值半群的原点切锥与 Hilbert 级数，证明除对称类外各拼接类均为 Cohen–Macaulay，而对称类满足 Rossi 猜想。
On ideals of affine semigroups and affine semigroups with maximal embedding dimension
本文刻画了由半群理想构造的I(S)-半群及同时为C-半群的情形，给出计算满足性质的I(S)-半群的算法，并引入和刻画了具有最大嵌入维数的仿射半群。
In their paper on the embeddings of the line in the plane, Abhyankar and Moh proved an important inequality, now known as the Abhyankar-Moh inequality, which can be stated in terms of the semigroup associated with the branch at infinity of a plane algebraic curve. Barrolleta, García Barroso and P\loski studied the semigroups of integers satisfying the Abhyankar-Moh inequality and call them Abhyankar-Moh semigroups. They described such semigroups with the maximum conductor. In this paper we prove that all possible conductor values are achieved for the Abhyankar-Moh semigroups of even degree. Ou…
The structure theorem for sets of length for numerical semigroups
本文针对任意数值半群证明了特化版长度集结构定理，利用由原半群生成元导出的两个数值半群的Apéry集及其间隙刻画大元素长度集的缺失长度，并给出满足定理的元素的明确界值。
Given two semigroups $\langle A\rangle$ and $\langle B\rangle$ in ${\mathbb N}^n$, we wonder when they can be glued, i.e., when there exists a semigroup $\langle C\rangle$ in ${\mathbb N}^n$ such that the defining ideals of the corresponding semigroup rings satisfy that $I_C=I_A+I_B+\langleρ\rangle$ for some binomial $ρ$. If $n\geq 2$ and $k[A]$ and $k[B]$ are Cohen-Macaulay, we prove that in order to glue them, one of the two semigroups must be degenerate. Then we study the two most degenerate cases: when one of the semigroups is generated by one single element (simple split) and the case whe…
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