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Galois groups of random polynomials over the rational function field
IF 1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-12-23 DOI: 10.1112/jlms.70061
Alexei Entin

For a fixed prime power q$q$ and natural number d$d$, we consider a random polynomial

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引用次数: 0
Groups acting on veering pairs and Kleinian groups
IF 1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-12-23 DOI: 10.1112/jlms.70052
Hyungryul Baik, Hongtaek Jung, KyeongRo Kim

We show that some subgroups of the orientation-preserving circle homeomorphism group with invariant veering pairs of laminations are hyperbolic 3-orbifold groups. On the way, we show that from a veering pair of laminations, one can construct a loom space (in the sense of Schleimer–Segerman) as a quotient. Our approach does not assume the existence of any 3-manifold to begin with, so this is a geometrization-type result, and supersedes some of the results regarding the relation among veering triangulations, pseudo-Anosov flows, taut foliations in the literature.

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引用次数: 0
The Poincaré-extended a b $mathbf {a}mathbf {b}$ -index 庞加莱姆-扩展了一个b $mathbf {a}mathbf {b}$ -索引
IF 1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-12-20 DOI: 10.1112/jlms.70054
Galen Dorpalen-Barry, Joshua Maglione, Christian Stump

Motivated by a conjecture concerning Igusa local zeta functions for intersection posets of hyperplane arrangements, we introduce and study the Poincaré-extended ab$mathbf {a}mathbf {b}$-index, which generalizes both the ab$mathbf {a}mathbf {b}$-index and the Poincaré polynomial. For posets admitting R$R$-labelings, we give a combinatorial description of the coefficients of the extended ab$mathbf {a}mathbf {b}$-index, proving their nonnegativity. In the case of intersection posets of hyperplane arrangements, we prove the above conjecture of the second author and Voll as well as another conjecture of the second author and Kühne. We also define the pullback ab$mathbf {a}mathbf {b}$-index, generalizing the cd$mathbf {c}mathbf {d}$-index of face posets for oriented matroids. Our results recover, generalize, and unify results from Billera–Ehrenborg–Readdy, Bergeron–Mykytiuk–Sottile–van Willigenburg, Saliola–Thomas, and Ehrenborg. This connection allows us to translate our results into the language of quasisymmetric functions, and — in the special case of symmetric functions — pose a conjecture about Schur positivity. This conjecture was strengthened and proved by Ricky Liu, and the proof appears as an appendix.

受一个关于超平面排列的交点集合的伊古萨局部zeta函数的猜想的启发,我们引入并研究了波恩卡莱扩展的a b $mathbf {a}mathbf {b}$ -指数,它概括了a b $mathbf {a}mathbf {b}$ -指数和波恩卡莱多项式。对于允许 R $R$ 标记的正集,我们给出了扩展的 a b $mathbf {a}mathbf {b}$ 指数系数的组合描述,并证明了它们的非负性。在超平面排列的交集正集情况下,我们证明了第二作者和沃尔的上述猜想,以及第二作者和库内的另一个猜想。我们还定义了回拉 a b $mathbf {a}mathbf {b}$ 索引,概括了定向矩阵的面正集的 c d $mathbf {c}mathbf {d}$ 索引。我们的结果恢复、概括并统一了比尔拉-艾伦伯格-雷迪、贝格龙-米基蒂乌克-索蒂莱-范-威利根堡、萨利奥拉-托马斯和艾伦伯格的结果。这种联系使我们能够将我们的结果转化为准对称函数的语言,并在对称函数的特殊情况下,提出了关于舒尔正定性的猜想。这个猜想得到了刘力奇的加强和证明,证明作为附录出现。
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引用次数: 0
The Ekström–Persson conjecture regarding random covering sets 关于随机覆盖集的Ekström-Persson猜想
IF 1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-12-20 DOI: 10.1112/jlms.70058
Esa Järvenpää, Maarit Järvenpää, Markus Myllyoja, Örjan Stenflo

We consider the Hausdorff dimension of random covering sets formed by balls with centres chosen independently at random according to an arbitrary Borel probability measure on Rd$mathbb {R}^d$ and radii given by a deterministic sequence tending to zero. We prove, for a certain parameter range, the conjecture by Ekström and Persson concerning the exact value of the dimension in the special case of radii (nα)n=1$(n^{-alpha })_{n=1}^infty$. For balls with an arbitrary sequence of radii, we find sharp bounds for the dimension and show that the natural extension of the Ekström–Persson conjecture is not true in this case. Finally, we construct examples demonstrating that there does not exist a dimension formula involving only the lower and upper local dimensions of the measure and a critical parameter determined by the sequence of radii.

我们考虑了随机覆盖集的Hausdorff维数,这些随机覆盖集是由球组成的,这些球的中心是根据R d $mathbb {R}^d$上的任意Borel概率度量随机选择的,半径由趋于零的确定性序列给定。我们证明,对于一定的参数范围,Ekström和Persson关于在半径(n−α) n =的特殊情况下维数精确值的猜想1∞$(n^{-alpha })_{n=1}^infty$。对于具有任意半径序列的球,我们找到了尺寸的明确界限,并证明了Ekström-Persson猜想的自然扩展在这种情况下是不成立的。最后,我们构造了实例,证明不存在只涉及测度的上下局部维数和由半径序列决定的关键参数的维数公式。
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引用次数: 0
Central limit theorem for smooth statistics of one-dimensional free fermions 一维自由费米子光滑统计量的中心极限定理
IF 1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-12-20 DOI: 10.1112/jlms.70045
Alix Deleporte, Gaultier Lambert

We consider the determinantal point processes associated with the spectral projectors of a Schrödinger operator on R$mathbb {R}$, with a smooth confining potential. In the semiclassical limit, where the number of particles tends to infinity, we obtain a Szegő-type central limit theorem for the fluctuations of smooth linear statistics. More precisely, the Laplace transform of any statistic converges without renormalisation to a Gaussian limit with a H1/2$H^{1/2}$-type variance, which depends on the potential. In the one-well (one-cut) case, using the quantum action-angle theorem and additional micro-local tools, we reduce the problem to the asymptotics of Fredholm determinants of certain approximately Toeplitz operators. In the multi-cut case, we show that for generic potentials, a similar result holds and the contributions of the different wells are independent in the limit.

我们考虑的是与 R $mathbb {R}$ 上的薛定谔算子谱投影相关的行列式点过程,它具有平滑的约束势。在粒子数趋于无穷大的半经典极限中,我们得到了平稳线性统计波动的塞格型中心极限定理。更准确地说,任何统计量的拉普拉斯变换都会在不进行重正化的情况下收敛到高斯极限,其方差为 H 1 / 2 $H^{1/2}$ 型,这取决于势能。在单阱(单切)情况下,利用量子作用角定理和额外的微局域工具,我们将问题简化为某些近似托普利兹算子的弗雷德霍姆行列式的渐近性。在多切情况下,我们证明了对于一般势,类似的结果成立,并且不同井的贡献在极限中是独立的。
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引用次数: 0
Normalizers and centralizers of subnormal subsystems of fusion systems 融合系统次正态子系统的归一化器和中心化器
IF 1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-12-18 DOI: 10.1112/jlms.70048
Ellen Henke

Every saturated fusion system corresponds to a group-like structure called a regular locality. In this paper we study (suitably defined) normalizers and centralizers of partial subnormal subgroups of regular localities. This leads to a reasonable notion of normalizers and centralizers of subnormal subsystems of fusion systems.

每一个饱和核聚变系统都对应于一个称为规则局域的类群结构。本文研究了正则位置的部分次正规子群的正化器和中心化器。这导致了一个合理的概念,即融合系统的次正规子系统的归一化器和中心化器。
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引用次数: 0
Asymptotic dimension for covers with controlled growth 有控制增长的封面的渐近维度
IF 1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-12-17 DOI: 10.1112/jlms.70043
David Hume, John M. Mackay, Romain Tessera
<p>We prove various obstructions to the existence of regular maps (or coarse embeddings) between commonly studied spaces. For instance, there is no regular map (or coarse embedding) <span></span><math> <semantics> <mrow> <msup> <mi>H</mi> <mi>n</mi> </msup> <mo>→</mo> <msup> <mi>H</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>×</mo> <mi>Y</mi> </mrow> <annotation>$mathbb {H}^nrightarrow mathbb {H}^{n-1}times Y$</annotation> </semantics></math> for <span></span><math> <semantics> <mrow> <mi>n</mi> <mo>⩾</mo> <mn>3</mn> </mrow> <annotation>$ngeqslant 3$</annotation> </semantics></math>, or <span></span><math> <semantics> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>T</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mi>n</mi> </msup> <mo>→</mo> <msup> <mrow> <mo>(</mo> <msub> <mi>T</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>×</mo> <mi>Y</mi> </mrow> <annotation>$(T_3)^n rightarrow (T_3)^{n-1}times Y$</annotation> </semantics></math> whenever <span></span><math> <semantics> <mi>Y</mi> <annotation>$Y$</annotation> </semantics></math> is a bounded degree graph with subexponential growth, where <span></span><math> <semantics> <msub> <mi>T</mi> <mn>3</mn> </msub> <annotation>$T_3$</annotation> </semantics></math> is the 3-regular tree. We also resolve Question 5.2 (<i>Groups Geom. Dyn</i>. <b>6</b> (2012), no. 4, 639–658), prov
我们证明了在通常研究的空间之间存在规则映射(或粗嵌入)的各种障碍。例如,没有规则映射(或粗嵌入)H n→H n−1 × Y $mathbb {H}^nrightarrow mathbb {H}^{n-1}times Y$对于n小于3 $ngeqslant 3$,或者(t3) n→(T3) n−1 × Y $(T_3)^n rightarrow (T_3)^{n-1}times Y$当Y $Y$是次指数增长的有界度图时,t3 $T_3$是三规则树。我们还解决了问题5.2(分组)。文献6 (2012),no. 6;4,639 - 658);证明当Y $Y$是有界时,不存在正则映射h2→t3 × Y $mathbb {H}^2 rightarrow T_3 times Y$当Y $Y$有次指数增长时,不存在拟等距嵌入。最后,我们证明了不存在正则映射F n→Z∶F n−1 $F^nrightarrow mathbb {Z}wr F^{n-1}$其中F $F$是两个生成器上的自由群。为了证明这些结果,我们引入并研究了允许无界覆盖控制增长的渐近维的推广。
{"title":"Asymptotic dimension for covers with controlled growth","authors":"David Hume,&nbsp;John M. Mackay,&nbsp;Romain Tessera","doi":"10.1112/jlms.70043","DOIUrl":"https://doi.org/10.1112/jlms.70043","url":null,"abstract":"&lt;p&gt;We prove various obstructions to the existence of regular maps (or coarse embeddings) between commonly studied spaces. For instance, there is no regular map (or coarse embedding) &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;→&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 &lt;mi&gt;Y&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathbb {H}^nrightarrow mathbb {H}^{n-1}times Y$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; for &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;⩾&lt;/mo&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$ngeqslant 3$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, or &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;→&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 &lt;mi&gt;Y&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$(T_3)^n rightarrow (T_3)^{n-1}times Y$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; whenever &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;Y&lt;/mi&gt;\u0000 &lt;annotation&gt;$Y$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a bounded degree graph with subexponential growth, where &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;T&lt;/mi&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$T_3$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is the 3-regular tree. We also resolve Question 5.2 (&lt;i&gt;Groups Geom. Dyn&lt;/i&gt;. &lt;b&gt;6&lt;/b&gt; (2012), no. 4, 639–658), prov","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70043","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142861831","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Modular representations of the Yangian Y 2 $Y_2$ Yangian y2 $Y_2$的模表示
IF 1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-12-16 DOI: 10.1112/jlms.70056
Hao Chang, Jinxin Hu, Lewis Topley

Let Y2$Y_2$ be the Yangian associated to the general linear Lie algebra gl2$mathfrak {gl}_2$, defined over an algebraically closed field k$mathbb {k}$ of characteristic p>0$p>0$. In this paper, we study the representation theory of the restricted Yangian Y2[p]$Y^{[p]}_2$. This leads to a description of the representations of gl2n$mathfrak {gl}_{2n}$, whose p$p$-character is nilpotent with Jordan type given by a two-row partition (n,n)$(n, n)$.

设y2 $Y_2$是与一般线性李代数gl2 $mathfrak {gl}_2$相关的Yangian,定义在特征p >的代数闭域k $mathbb {k}$上;0$ p>0$。本文研究了受限Yangian Y 2 [p] $Y^{[p]}_2$的表示理论。这导致了对gl 2n $mathfrak {gl}_{2n}$表示的描述,其中p$ p$ -字符是幂零的,具有由两行划分(n, n)$ (n, n)$给出的Jordan类型。
{"title":"Modular representations of the Yangian \u0000 \u0000 \u0000 Y\u0000 2\u0000 \u0000 $Y_2$","authors":"Hao Chang,&nbsp;Jinxin Hu,&nbsp;Lewis Topley","doi":"10.1112/jlms.70056","DOIUrl":"https://doi.org/10.1112/jlms.70056","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Y</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$Y_2$</annotation>\u0000 </semantics></math> be the Yangian associated to the general linear Lie algebra <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>gl</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$mathfrak {gl}_2$</annotation>\u0000 </semantics></math>, defined over an algebraically closed field <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$mathbb {k}$</annotation>\u0000 </semantics></math> of characteristic <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>&gt;</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$p&gt;0$</annotation>\u0000 </semantics></math>. In this paper, we study the representation theory of the restricted Yangian <span></span><math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>Y</mi>\u0000 <mn>2</mn>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mi>p</mi>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 </msubsup>\u0000 <annotation>$Y^{[p]}_2$</annotation>\u0000 </semantics></math>. This leads to a description of the representations of <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>gl</mi>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>$mathfrak {gl}_{2n}$</annotation>\u0000 </semantics></math>, whose <span></span><math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math>-character is nilpotent with Jordan type given by a two-row partition <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>,</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(n, n)$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70056","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142861583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Fuchs' problem for finitely generated abelian groups: The small torsion case 有限生成阿贝尔群的Fuchs问题:小扭转情况
IF 1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-12-16 DOI: 10.1112/jlms.70055
I. Del Corso, L. Stefanello

A classical problem, raised by Fuchs in 1960, asks to classify the abelian groups which are groups of units of some rings. In this paper, we consider the case of finitely generated abelian groups, solving Fuchs' problem for such groups with the additional assumption that the torsion subgroups are small, for a suitable notion of small related to the Prüfer rank. As a concrete instance, we classify for each n2$ngeqslant 2$ the realisable groups of the form Z/nZ×Zr$mathbb {Z}/nmathbb {Z}times mathbb {Z}^r$. Our tools require an investigation of the adjoint group of suitable radical rings of odd prime power order appearing in the picture, giving conditions under which the additive and adjoint groups are isomorphic. In the last section, we also deal with some groups of order a power of 2, proving that the groups of the form Z/4Z×Z/2uZ$mathbb {Z}/4mathbb {Z}times mathbb {Z}/2^{u}mathbb {Z}$ are realisable if and only if 0u3$0leqslant uleqslant 3$ or 2u+1$2^u+1$ is a Fermat prime.

福克斯在 1960 年提出了一个经典问题,要求对作为某些环的单位群的无常群进行分类。在本文中,我们考虑了有限生成的无边群的情况,在解决福克斯的问题时,附加了一个假设,即对于与普吕弗秩相关的合适的小概念,扭转子群是小的。作为一个具体的例子,我们对每个 n ⩾ 2 $ngeqslant 2$ 形式为 Z / n Z × Z r $mathbb {Z}/nmathbb {Z}times mathbb {Z}^r$ 的可实现群进行了分类。我们的工具要求研究图中出现的奇素数幂阶的合适基环的邻接群,给出加群和邻接群同构的条件。在最后一节中,我们还将讨论一些阶为 2 的幂的群、证明当且仅当 0 ⩽ u ⩽ 3 $0leqslant uleqslant 3$ 或 2 u + 1 $2^u+1$ 是费马素数时,形式为 Z / 4 Z × Z / 2 u Z $mathbb {Z}/4mathbb {Z}/times mathbb {Z}/2^{u}mathbb {Z}$ 的群是可实现的。
{"title":"On Fuchs' problem for finitely generated abelian groups: The small torsion case","authors":"I. Del Corso,&nbsp;L. Stefanello","doi":"10.1112/jlms.70055","DOIUrl":"https://doi.org/10.1112/jlms.70055","url":null,"abstract":"<p>A classical problem, raised by Fuchs in 1960, asks to classify the abelian groups which are groups of units of some rings. In this paper, we consider the case of finitely generated abelian groups, solving Fuchs' problem for such groups with the additional assumption that the torsion subgroups are <i>small</i>, for a suitable notion of small related to the Prüfer rank. As a concrete instance, we classify for each <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>⩾</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$ngeqslant 2$</annotation>\u0000 </semantics></math> the realisable groups of the form <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Z</mi>\u0000 <mo>/</mo>\u0000 <mi>n</mi>\u0000 <mi>Z</mi>\u0000 <mo>×</mo>\u0000 <msup>\u0000 <mi>Z</mi>\u0000 <mi>r</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$mathbb {Z}/nmathbb {Z}times mathbb {Z}^r$</annotation>\u0000 </semantics></math>. Our tools require an investigation of the adjoint group of suitable radical rings of odd prime power order appearing in the picture, giving conditions under which the additive and adjoint groups are isomorphic. In the last section, we also deal with some groups of order a power of 2, proving that the groups of the form <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Z</mi>\u0000 <mo>/</mo>\u0000 <mn>4</mn>\u0000 <mi>Z</mi>\u0000 <mo>×</mo>\u0000 <mi>Z</mi>\u0000 <mo>/</mo>\u0000 <msup>\u0000 <mn>2</mn>\u0000 <mi>u</mi>\u0000 </msup>\u0000 <mi>Z</mi>\u0000 </mrow>\u0000 <annotation>$mathbb {Z}/4mathbb {Z}times mathbb {Z}/2^{u}mathbb {Z}$</annotation>\u0000 </semantics></math> are realisable if and only if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>0</mn>\u0000 <mo>⩽</mo>\u0000 <mi>u</mi>\u0000 <mo>⩽</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$0leqslant uleqslant 3$</annotation>\u0000 </semantics></math> or <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mn>2</mn>\u0000 <mi>u</mi>\u0000 </msup>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$2^u+1$</annotation>\u0000 </semantics></math> is a Fermat prime.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70055","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142861584","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Results on left–right Artin approximation for algebraic morphisms and for analytic morphisms of weakly-finite singularity type 代数态射和弱有限奇异型解析态射的左右Artin近似结果
IF 1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-12-16 DOI: 10.1112/jlms.70053
Dmitry Kerner
<p>The classical Artin approximation (AP) reads: any formal solution of a system of (analytic, resp., algebraic) equations of implicit function type is approximated by “ordinary” solutions (i.e., analytic, resp., algebraic). Morphisms of scheme-germs, for example, <span></span><math> <semantics> <mrow> <mi>M</mi> <mi>a</mi> <mi>p</mi> <mi>s</mi> <mo>(</mo> <mrow> <mo>(</mo> <msup> <mi>k</mi> <mi>n</mi> </msup> <mo>,</mo> <mi>o</mi> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <msup> <mi>k</mi> <mi>m</mi> </msup> <mo>,</mo> <mi>o</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <annotation>$Mapsbig ((mathbb {k}^n,o),(mathbb {k}^m,o)big)$</annotation> </semantics></math>, are usually studied up to the left–right equivalence. The natural question is the left–right version of AP: when is the formal left–right equivalence of morphisms approximated by the “ordinary” (i.e., analytic, resp., algebraic) equivalence? In this case, the standard AP is not directly applicable, as the involved (functional) equations are not of implicit function type. Moreover, the naive extension does not hold in the analytic case, because of Osgood–Gabrielov–Shiota examples. The left–right version of Artin approximation (<span></span><math> <semantics> <mrow> <mi>L</mi> <mi>R</mi> </mrow> <annotation>$mathcal Lmathcal {R}$</annotation> </semantics></math>.AP) was established by M. Shiota for morphisms that are either Nash or [real-analytic and of finite singularity type]. We establish <span></span><math> <semantics> <mrow> <mi>L</mi> <mi>R</mi> </mrow> <annotation>$mathcal Lmathcal {R}$</annotation> </semantics></math>.AP and its stronger version of Płoski (<span></span><math> <semantics> <mrow> <mi>L</mi> <mi>R</mi> </mrow> <annotation>$mathcal Lmathcal {R}$</annotation> </semantics></math>.APP) for <span></span><math> <semantics> <mrow> <mi>M</mi> <mi>a</mi> <mi>p</mi> <mi>s</mi> <mo>(</mo> <mi>X</mi> <mo>,</mo>
经典的阿尔丁近似(AP)是这样的:隐函数式(解析式,或代数式)方程组的任何形式解都可以用 "普通 "解(即解析式,或代数式)来近似。图解符号的变形,例如 M a p s ( ( k n , o ) , ( k m , o ) ) , ( k m , o ) ) $Mapsbig ((mathbb {k}^n,o),(mathbb {k}^m,o)big)$ ,通常研究到左右等价为止。自然而然的问题是左-右版本的 AP:什么时候形态的形式左-右等价近似于 "普通"(即解析的,也就是代数的)等价?在这种情况下,标准 AP 并不直接适用,因为所涉及的(函数)方程并不属于隐函数类型。此外,由于 Osgood-Gabrielov-Shiota 例子,天真的扩展在解析情况下也不成立。汐田(M. Shiota)针对纳什或[实解析和有限奇点类型]的形态建立了阿廷近似的左右版本(L R $mathcal Lmathcal {R}$ .AP)。我们为 M a p s ( X , Y ) $Maps (X,Y)$ 建立了 L R $mathcal Lmathcal {R}$ .AP 及其更强版本 Płoski ( L R $mathcal Lmathcal {R}$ .APP) ,其中 X , Y $X,Y$ 是任意特征方案的解析/代数胚芽。更确切地说: 后一类 "弱有限奇点类型 "的态射(我们引入的)具有单独的重要性。它自然地扩展了传统的 "有限奇异性类型 "变形,同时保留了它们的非病理性行为。这个定义是通过具有奇异目标的态的高临界位置和高判别式来实现的。我们建立了这些临界点的基本性质。特别是,任何映射都是由其高临界点有限(右)决定的。
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Journal of the London Mathematical Society-Second Series
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