Journal of the London Mathematical Society-Second Series最新文献

{"title":"Galois groups of random polynomials over the rational function field","authors":"Alexei Entin","doi":"10.1112/jlms.70061","DOIUrl":"https://doi.org/10.1112/jlms.70061","url":null,"abstract":"<p>For a fixed prime power <span></span><math>\u0000 <semantics>\u0000 <mi>q</mi>\u0000 <annotation>$q$</annotation>\u0000 </semantics></math> and natural number <span></span><math>\u0000 <semantics>\u0000 <mi>d</mi>\u0000 <annotation>$d$</annotation>\u0000 </semantics></math>, we consider a random polynomial\u0000\u0000 </p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143118215","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Groups acting on veering pairs and Kleinian groups","authors":"Hyungryul Baik,&nbsp;Hongtaek Jung,&nbsp;KyeongRo Kim","doi":"10.1112/jlms.70052","DOIUrl":"https://doi.org/10.1112/jlms.70052","url":null,"abstract":"<p>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.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143118216","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Poincaré-extended \u0000 \u0000 \u0000 a\u0000 b\u0000 \u0000 $mathbf {a}mathbf {b}$\u0000 -index","authors":"Galen Dorpalen-Barry,&nbsp;Joshua Maglione,&nbsp;Christian Stump","doi":"10.1112/jlms.70054","DOIUrl":"https://doi.org/10.1112/jlms.70054","url":null,"abstract":"<p>Motivated by a conjecture concerning Igusa local zeta functions for intersection posets of hyperplane arrangements, we introduce and study the <i>Poincaré-extended</i> <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mi>b</mi>\u0000 </mrow>\u0000 <annotation>$mathbf {a}mathbf {b}$</annotation>\u0000 </semantics></math><i>-index</i>, which generalizes both the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mi>b</mi>\u0000 </mrow>\u0000 <annotation>$mathbf {a}mathbf {b}$</annotation>\u0000 </semantics></math>-index and the Poincaré polynomial. For posets admitting <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$R$</annotation>\u0000 </semantics></math>-labelings, we give a combinatorial description of the coefficients of the extended <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mi>b</mi>\u0000 </mrow>\u0000 <annotation>$mathbf {a}mathbf {b}$</annotation>\u0000 </semantics></math>-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 <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mi>b</mi>\u0000 </mrow>\u0000 <annotation>$mathbf {a}mathbf {b}$</annotation>\u0000 </semantics></math>-index, generalizing the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>c</mi>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation>$mathbf {c}mathbf {d}$</annotation>\u0000 </semantics></math>-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.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70054","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142868871","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}

{"title":"The Ekström–Persson conjecture regarding random covering sets","authors":"Esa Järvenpää,&nbsp;Maarit Järvenpää,&nbsp;Markus Myllyoja,&nbsp;Örjan Stenflo","doi":"10.1112/jlms.70058","DOIUrl":"https://doi.org/10.1112/jlms.70058","url":null,"abstract":"<p>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 <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>d</mi>\u0000 </msup>\u0000 <annotation>$mathbb {R}^d$</annotation>\u0000 </semantics></math> 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 <span></span><math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>n</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mi>α</mi>\u0000 </mrow>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>=</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <mi>∞</mi>\u0000 </msubsup>\u0000 <annotation>$(n^{-alpha })_{n=1}^infty$</annotation>\u0000 </semantics></math>. 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.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70058","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142868872","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}

{"title":"Central limit theorem for smooth statistics of one-dimensional free fermions","authors":"Alix Deleporte,&nbsp;Gaultier Lambert","doi":"10.1112/jlms.70045","DOIUrl":"https://doi.org/10.1112/jlms.70045","url":null,"abstract":"<p>We consider the determinantal point processes associated with the spectral projectors of a Schrödinger operator on <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$mathbb {R}$</annotation>\u0000 </semantics></math>, 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 <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>H</mi>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>/</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </msup>\u0000 <annotation>$H^{1/2}$</annotation>\u0000 </semantics></math>-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.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70045","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142868975","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}

{"title":"Normalizers and centralizers of subnormal subsystems of fusion systems","authors":"Ellen Henke","doi":"10.1112/jlms.70048","DOIUrl":"https://doi.org/10.1112/jlms.70048","url":null,"abstract":"<p>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.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70048","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142861446","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}

{"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}

{"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}

{"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}

{"title":"Results on left–right Artin approximation for algebraic morphisms and for analytic morphisms of weakly-finite singularity type","authors":"Dmitry Kerner","doi":"10.1112/jlms.70053","DOIUrl":"https://doi.org/10.1112/jlms.70053","url":null,"abstract":"&lt;p&gt;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, &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;o&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;k&lt;/mi&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;o&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$Mapsbig ((mathbb {k}^n,o),(mathbb {k}^m,o)big)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, 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 (&lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathcal Lmathcal {R}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;.AP) was established by M. Shiota for morphisms that are either Nash or [real-analytic and of finite singularity type]. We establish &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathcal Lmathcal {R}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;.AP and its stronger version of Płoski (&lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathcal Lmathcal {R}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;.APP) for &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 ","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":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142861492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}