Ayomikun Adeniran, Steve Butler, Colin Defant, Yibo Gao, Pamela E. Harris, Cyrus Hettle, Qingzhong Liang, Hayan Nam, Adam Volk
We find a relation between the genus of a quotient of a numerical semigroup $S$ and the genus of $S$ itself. We use this identity to compute the genus of a quotient of $S$ when $S$ has embedding dimension $2$. We also exhibit identities relating the Frobenius numbers and the genus of quotients of numerical semigroups that are generated by certain types of arithmetic progressions.
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In this paper we present a new approach to construct the set of numerical semigroups with a fixed genus. Our methodology is based on the construction of the set of numerical semigroups with fixed Frobenius number and genus. An equivalence relation is given over this set and a tree structure is defined for each equivalence class. We also provide a more efficient algorithm based in the translation of a numerical semigroup to its so-called Kunz-coordinates vector.
We conjecture a Fibonacci-like property on the number of numerical semigroups of a given genus. Moreover we conjecture that the associated quotient sequence approaches the golden ratio. The conjecture is motivated by the results on the number of semigroups of genus at most 50. The Wilf conjecture has also been checked for all numerical semigroups with genus in the same range.
On quotients of numerical semigroups for almost arithmetic progressions
研究数值半群商的性质，通过将Apéry集的计算转化为简单优化问题，推导出某些特殊半群的Frobenius数闭式公式，解决了一个开放性问题。
New Eliahou Semigroups and Verification of the Wilf Conjecture for Genus up to 65
该研究通过图形化重释种子算法探索数值半群树，找出亏格≤65的所有Eliahou半群，验证了这些半群满足Wilf猜想，证明该猜想在亏格65以内成立。
Rarity of the infinite chains in the tree of numerical semigroups
本研究解决了2009年提出的公开问题，证明随亏格趋于无穷，固定亏格下属于数值半群树无限链的半群占比趋于0，即多数数值半群在树中仅有有限后代。
Let $\mathcal{S}$ be a finite commutative semigroup. The Davenport constant of $\mathcal{S}$, denoted $D(\mathcal{S})$, is defined to be the least positive integer $d$ such that every sequence $T$ of elements in $\mathcal{S}$ of length at least $d$ contains a subsequence $T'$ with the sum of all terms from $T'$ equaling the sum of all terms from $T$. Let $\F_p[x]$ be a polynomial ring in one variable over the prime field $\F_p$, and let $f(x)\in \F_p[x]$. In this paper, we made a study of the Davenport constant of the multiplicative semigroup of the quotient ring $\frac{\F_p[x]}{\langle f(x)\r…
A numerical semigroup is said to be ordinary if it has all its gaps in a row. Indeed, it contains zero and all integers from a given positive one. One can define a simple operation on a non-ordinary semigroup, which we call here the ordinarization transform, by removing its smallest non-zero non-gap (the multiplicity) and adding its largest gap (the Frobenius number). This gives another numerical semigroup and by repeating this transform several times we end up with an ordinary semigroup. The genus, that is, the number of gaps, is kept constant in all the transforms. This procedure allows the …
On the Frobenius Number and Genus of a Collection of Semigroups Generalizing Repunit Numerical Semigroups
本文研究了特定数数半群的弗罗贝尼乌斯数和亏格，提出d可为负整数的新方法，并针对特殊情形如梅森、塔比特和雷努特数数半群给出简化公式，部分解决了普罗思数数半群的开放问题。
Let $n_g$ be the number of numerical semigroups of genus $g$. We present an approach to compute $n_g$ by using even gaps, and the question: Is it true that $n_{g+1}>n_g$? is investigated. Let $N_γ(g)$ be the number of numerical semigroups of genus $g$ whose number of even gaps equals $γ$. We show that $N_γ(g)=N_γ(3γ)$ for $γ\leq \lfloor g/3\rfloor$ and $N_γ(g)=0$ for $γ> \lfloor 2g/3\rfloor$; thus the question above is true provided that $N_γ(g+1) > N_γ(g)$ for $γ= \lfloor g/3 \rfloor +1, \ldots, \lfloor 2g/3\rfloor$. We also show that $N_γ(3γ)$ coincides with $f_γ$, the number introduced by B…
In this article, we study the quotients of numerical semigroups, generated by two coprime positive numbers, named (a,b) over d. We give formulae for the usual invariants of these semigroups, expressed in terms of continued fraction expansions and Ostrowski-like numeration of some rationals, simply related to entries a, b, d. So, we obtain quadratic complexity algorithms to compute these invariants. As a consequence, we will prove that, for these numerical semigroups, the type is always lower than the embedding dimension and deduce Wilf's property. We also consider the '' reverse problem'' : gi…
On the quotient of affine semigroups by a positive integer
本研究探讨了仿射半群被正整数除的商的性质，揭示了关联树工具，推广了关键不可约性结果至$\mathcal C$-半群商类，并引入了仿射半群的算术簇新概念。
Given three pairwise coprime positive integers $a_1,a_2,a_3 \in \mathbb{Z}^+$ we show the existence of a relation between the sets of the first elements of the three quotients $\frac{\langle a_i,a_j \rangle}{a_k}$ that can be made for every $\{i.j,k\}=\{1,2,3\}$. Then we use this result to give an improved version of Johnson's semi-explicit formula for the Frobenius number $g(a_1,a_2,a_3)$ without restriction on the choice of $a_1,a_2,a_3$ and to give an explicit formula for a particular class of numerical semigroups.
We define a reflective numerical semigroup of genus $g$ as a numerical semigroup that has a certain reflective symmetry when viewed within $\mathbb{Z}$ as an array with $g$ columns. Equivalently, a reflective numerical semigroup has one gap in each residue class modulo $g$. In this paper, we give an explicit description for all reflective numerical semigroups. With this, we can describe the reflective members of well-known families of numerical semigroups as well as obtain formulas for the number of reflective numerical semigroups of a given genus or given Frobenius number.
Given two numerical semigroups $S$ and $T$ and a positive integer $d$, $S$ is said to be one over $d$ of $T$ if $S=\{s \in \mathbb{N} \ | \ ds \in T \}$ and in this case $T$ is called a $d$-fold of $S$. We prove that the minimal genus of the $d$-folds of $S$ is $g + \lceil \frac{(d-1)f}{2} \rceil$, where $g$ and $f$ denote the genus and the Frobenius number of $S$. The case $d=2$ is a problem proposed by Robles-Pérez, Rosales, and Vasco. Furthermore, we find the minimal genus of the symmetric doubles of $S$ and study the particular case when $S$ is almost symmetric. Finally, we study the Frobe…
Shimura curve Atkin--Lehner quotients of genus at most two
本研究完全枚举了志村曲线$X_0^D(N)$在Atkin–Lehner对合子群作用下亏格≤2的商，确定了1270条亏格1商的雅可比同构类，给出了146条非椭圆亏格1曲线与405条亏格2商曲线的定义方程，推广了Čerednik–Drinfeld单值化理论的应用范围。
We use the method of quadratic Chabauty on the quotients $X_0^+(N)$ of modular curves $X_0(N)$ by their Fricke involutions to provably compute all the rational points of these curves for prime levels $N$ of genus four, five, and six. We find that the only such curves with exceptional rational points are of levels $137$ and $311$. In particular there are no exceptional rational points on those curves of genus five and six. More precisely, we determine the rational points on the curves $X_0^+(N)$ for $N=137,173,199,251,311,157,181,227,263,163,197,211,223,269,271,359$.
Weierstrass semigroups and automorphism group of a maximal curve with the third largest genus
本文确定具有第三大亏格的极大曲线$\mathcal{X}_3$在所有点处的Weierstrass半群与完整自同构群，揭示其Weierstrass点异常丰富，并证明除小$q$外其自同构群完全继承自Hermitian曲线，为唯一性公开问题提供关键线索。
Let $\F_q[x]$ be the ring of polynomials over the finite field $\F_q$, and let $f$ be a polynomial of $\F_q[x]$. Let $R=\frac{\F_q[x]}{(f)}$ be a quotient ring of $\F_q[x]$ with $0\neq R\neq \F_q[x]$. Let $\mathcal{S}_R$ be the multiplicative semigroup of the ring $R$, and let ${\rm U}(\mathcal{S}_R)$ be the group of units of $\mathcal{S}_R$. The Davenport constant ${\rm D}(\mathcal{S}_R)$ of the multiplicative semigroup $\mathcal{S}_R$ is the least positive integer $\ell$ such that for any $\ell$ polynomials $g_1,g_2,\ldots,g_{\ell}\in \F_q[x]$, there exists a subset $I\subsetneq [1,\ell]$ wi…
$p$-numerical semigroup of generalized Fibonacci triples
本文针对非负整数 $p$，给出了涉及 Jacobsthal 多项式（包含 Fibonacci 数）的广义 Fibonacci 数值半群的 $p$-Frobenius 数和 $p$-亏格显式公式，并探讨了其在 $p$-Hilbert 级数上的应用。
Semigroups from full lattices in commutative ${\mathbb Q}$-algebras
本文研究了交换有理代数中满格构成的半群及其商半群，扩展了乔丹-扎森豪斯定理，并探讨了其在正交群共轭类中的应用。
Representability of systems of proportionally modular numerical semigroups
研究比例模数值半群系统的可表示性问题，核心方法是从其商描述出发，通过双生成元数值半群实现每个商，再用双分支典范等变分解图及合适重数胶合，主要贡献是证明其可由特定加权齐次曲面奇点的分解表示。
Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, $q \equiv 1 \pmod 3$
针对 q≡1 mod 3 时具有第三大亏格的极大函数域 Y_3，本文显式确定其所有位的 Weierstrass 半群与完整自同构群，发现其 Weierstrass 位结构异常丰富，但自同构群除小 q 外均继承自 Hermitian 函数域。
Numerical semigroups are cofinite additive submonoids of the natural numbers. In 2011, Keith and Nath illustrated an injection from numerical semigroups to integer partitions. We explore this connection between partitions and numerical semigroups with a focus on classifying the partitions that appear in the image of the injection from numerical semigroups. In particular, we count the number of partitions that correspond to numerical semigroups in terms of genus, Frobenius number, and multiplicity, with some restrictions.
Eta quotients on $Γ_0(6)$ yield evaluations of sunrise integrals at 2, 3, 4 and 6 loops. At 2 and 3 loops, they provide modular parametrizations of inhomogeneous differential equations whose solutions are readily obtained by expanding in the nome $q$. Atkin-Lehner transformations that permute cusps ensure fast convergence for all external momenta. At 4 and 6 loops, on-shell integrals are periods of modular forms of weights 4 and 6 given by Eichler integrals of eta quotients. Weakly holomorphic eta quotients determine quasi-periods. A Rademacher sum formula is given for Fourier coefficients of …
We extend Lax-Phillips' theorem on discreteness of pseudo-cuspforms, in the style of Colin de Verdi{è}re's use of the Friedrichs self-adjoint extension of a restriction of the Laplace-Beltrami operator, as opposed to the use of semigroup methods. We use this to prove meromorphic continuation of Eisenstein series in several $\mathbb{Q}$-rank one cases, again following Colin de Verdi{è}re, as opposed to the semigroup-oriented viewpoint of Lax-Phillips and W. Mueller.
Rational Points of some genus $3$ curves from the rank $0$ quotient strategy
本文提出一种算法，用于计算亏格3曲线中作为秩0亏格1曲线二次覆盖情形的全部有理点，并在Magma中实现，在约4万条曲线上验证，找到了达到Stoll上界的实例。
Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, $q \equiv 0 \pmod 3$
本文针对 $q\equiv0\pmod3$，显式确定第三大亏格的 $\mathbb{F}_{q^2}$-极大函数域 $Z_3$ 的各位 Weierstrass 半群与完整自同构群。
Eichler orders, quotient graphs and random walks
本文研究如何由一个位处的 Bruhat–Tits 商图推断另一位处的商图，给出例外顶点集及使其为空的几何条件，并推广已有公式至其他序属与曲线。
Curves of genus 2 with group of automorphisms isomorphic to D_8 or D_12
本文对任意域k上的亏格为2的曲线进行了分类，其中的自同构群自同构群同构于D_8或D_12（在D_8情形下其特征不同于2，在D_{12}情形下其特征不同于2和3），直到k-同构。作为亏格2曲线分类的一个应用，我们得到了它们的椭圆曲线和雅可比曲线的精确算术信息。在域k=Q上，我们证明了具有自同构D_8和D_{12}的曲线的椭圆导数分别精确地是2次和3次Q曲线，并确定了哪些曲线具有GL_2型Jacobian。
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。
In this paper we study numerical semigroups containing a given positive integer and closed with respect to the action of an affine map. For such semigroups we find a minimal set of generators, their embedding dimension, their genus and their Frobenius number.
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