{"title":"The stable rank of Z[x] is 3","authors":"Luc Guyot","doi":"10.1016/j.exmath.2025.125659","DOIUrl":"10.1016/j.exmath.2025.125659","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>Z</mi><mrow><mo>[</mo><mi>x</mi><mo>]</mo></mrow></mrow></math></span> be the ring of univariate polynomials over <span><math><mi>Z</mi></math></span> and denote by <span><math><mrow><mo>sr</mo><mrow><mo>(</mo><mi>Z</mi><mrow><mo>[</mo><mi>x</mi><mo>]</mo></mrow><mo>)</mo></mrow></mrow></math></span> its stable rank in the sense of Bass. Grunewald, Mennicke and Vaserstein proved that <span><math><mrow><mo>sr</mo><mrow><mo>(</mo><mi>Z</mi><mrow><mo>[</mo><mi>x</mi><mo>]</mo></mrow><mo>)</mo></mrow><mo>=</mo><mn>3</mn><mo>.</mo></mrow></math></span> As the inequality <span><math><mrow><mo>sr</mo><mrow><mo>(</mo><mi>Z</mi><mrow><mo>[</mo><mi>x</mi><mo>]</mo></mrow><mo>)</mo></mrow><mo>≤</mo><mn>3</mn></mrow></math></span> follows immediately from Bass’s stable range theorem, the above identity is equivalent to the existence of a non-stable unimodular row of size 3. This note addresses minor errors found in the existing proof of the latter fact. Using the same methods, we show that the unimodular row <span><math><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>,</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>16</mn><mo>)</mo></mrow></math></span> is not stable.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 3","pages":"Article 125659"},"PeriodicalIF":0.8,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Representations of real numbers by alternating Perron series and their geometry","authors":"Mykola Moroz","doi":"10.1016/j.exmath.2024.125635","DOIUrl":"10.1016/j.exmath.2024.125635","url":null,"abstract":"<div><div>We consider the representation of real numbers by alternating Perron series (<span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>−</mo></mrow></msup></math></span>-representation), which is a generalization of representations of real numbers by Ostrogradsky–Sierpiński–Pierce series (Pierce series), alternating Sylvester series (second Ostrogradsky series), alternating Lüroth series, etc. Namely, we prove the basic topological and metric properties of <span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>−</mo></mrow></msup></math></span>-representation and find the relationship between <span><math><mi>P</mi></math></span>-representation and <span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>−</mo></mrow></msup></math></span>-representation in some measure theory problems.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 1","pages":"Article 125635"},"PeriodicalIF":0.8,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133974","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Extremal length and duality","authors":"Kai Rajala","doi":"10.1016/j.exmath.2024.125634","DOIUrl":"10.1016/j.exmath.2024.125634","url":null,"abstract":"<div><div>Classical extremal length (or conformal modulus) is a conformal invariant involving families of paths on the Riemann sphere. In “Extremal length and functional completion”, Fuglede initiated an abstract theory of extremal length which has since been widely applied. Concentrating on duality properties and applications to quasiconformal analysis, we demonstrate the flexibility of the theory and present recent advances in three different settings:</div><div>(1) Extremal length and uniformization of metric surfaces.</div><div>(2) Extremal length of families of surfaces and quasiconformal maps between <span><math><mi>n</mi></math></span>-dimensional spaces.</div><div>(3) Schramm’s transboundary extremal length and conformal maps between multiply connected plane domains.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 1","pages":"Article 125634"},"PeriodicalIF":0.8,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Conformal tilings, combinatorial curvature, and the type problem","authors":"Mohith Raju Nagaraju","doi":"10.1016/j.exmath.2024.125633","DOIUrl":"10.1016/j.exmath.2024.125633","url":null,"abstract":"<div><div>Roughly, a conformal tiling of a Riemann surface is a tiling where each tile is a suitable conformal image of a Euclidean regular polygon. In 1997, Bowers and Stephenson constructed an edge-to-edge conformal tiling of the complex plane using conformally regular pentagons. In contrast, we show that for all <span><math><mrow><mi>n</mi><mo>≥</mo><mn>7</mn></mrow></math></span>, there is no edge-to-edge conformal tiling of the complex plane using conformally regular <span><math><mi>n</mi></math></span>-gons. More generally, we discuss a relationship between the combinatorial curvature at each vertex of the conformal tiling and the universal cover (sphere, plane, or disc) of the underlying Riemann surface. This result follows from the work of Stone (1976) and Oh (2005) through a rich interplay between Riemannian geometry and combinatorial geometry. We provide an exposition of these proofs and some new applications to conformal tilings.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 1","pages":"Article 125633"},"PeriodicalIF":0.8,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143133973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Commutators and products of Lie ideals of prime rings","authors":"Tsiu-Kwen Lee , Jheng-Huei Lin","doi":"10.1016/j.exmath.2025.125658","DOIUrl":"10.1016/j.exmath.2025.125658","url":null,"abstract":"<div><div>Motivated by some recent results on Lie ideals, it is proved that if <span><math><mi>L</mi></math></span> is a Lie ideal of a simple ring <span><math><mi>R</mi></math></span> with center <span><math><mrow><mi>Z</mi><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>, then <span><math><mrow><mi>L</mi><mo>⊆</mo><mi>Z</mi><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>L</mi><mo>=</mo><mi>Z</mi><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mi>a</mi><mo>+</mo><mi>Z</mi><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> for some noncentral <span><math><mrow><mi>a</mi><mo>∈</mo><mi>L</mi></mrow></math></span>, or <span><math><mrow><mrow><mo>[</mo><mi>R</mi><mo>,</mo><mi>R</mi><mo>]</mo></mrow><mo>⊆</mo><mi>L</mi></mrow></math></span>, which gives a generalization of a classical theorem due to Herstein. We also study commutators and products of noncentral Lie ideals of prime rings. Precisely, let <span><math><mi>R</mi></math></span> be a prime ring with extended centroid <span><math><mi>C</mi></math></span>. We completely characterize Lie ideals <span><math><mi>L</mi></math></span> and elements <span><math><mi>a</mi></math></span> of <span><math><mi>R</mi></math></span> such that <span><math><mrow><mi>L</mi><mo>+</mo><mi>a</mi><mi>L</mi></mrow></math></span> contains a nonzero ideal of <span><math><mi>R</mi></math></span>. Given noncentral Lie ideals <span><math><mrow><mi>K</mi><mo>,</mo><mi>L</mi></mrow></math></span> of <span><math><mi>R</mi></math></span>, it is proved that <span><math><mrow><mrow><mo>[</mo><mi>K</mi><mo>,</mo><mi>L</mi><mo>]</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span> if and only if <span><math><mrow><mi>K</mi><mi>C</mi><mo>=</mo><mi>L</mi><mi>C</mi><mo>=</mo><mi>C</mi><mi>a</mi><mo>+</mo><mi>C</mi></mrow></math></span> for any noncentral element <span><math><mrow><mi>a</mi><mo>∈</mo><mi>L</mi></mrow></math></span>. As a consequence, we characterize noncentral Lie ideals <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math></span> with <span><math><mrow><mi>m</mi><mo>≥</mo><mn>2</mn></mrow></math></span> such that <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math></span> contains a nonzero ideal of <span><math><mi>R</mi></math></span>. Finally, we characterize noncentral Lie ideals <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>’s and <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>’s satisfying <span><math><mrow><mrow><mo>[</mo><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>K</mi></mr","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 3","pages":"Article 125658"},"PeriodicalIF":0.8,"publicationDate":"2025-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349059","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On moment functionals with signed representing measures","authors":"Konrad Schmüdgen","doi":"10.1016/j.exmath.2025.125657","DOIUrl":"10.1016/j.exmath.2025.125657","url":null,"abstract":"<div><div>Suppose that <span><math><mi>A</mi></math></span> is a finitely generated commutative unital real algebra and <span><math><mi>K</mi></math></span> is a closed subset of the set <span><math><mover><mrow><mi>A</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span> of characters of <span><math><mi>A</mi></math></span>. We study the following problem: When is <em>each</em> linear functional <span><math><mrow><mi>L</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>R</mi></mrow></math></span> an integral with respect to some signed Radon measure on <span><math><mover><mrow><mi>A</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span> supported by the set <span><math><mi>K</mi></math></span>? A complete characterization of these sets <span><math><mi>K</mi></math></span> and algebras <span><math><mi>A</mi></math></span> by necessary and sufficient conditions is given. The result is applied to the polynomial algebra <span><math><mrow><mi>R</mi><mrow><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span> and subsets <span><math><mi>K</mi></math></span> of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 3","pages":"Article 125657"},"PeriodicalIF":0.8,"publicationDate":"2025-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349099","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Characters of the unitriangular group and the Mackey method","authors":"Mikhail Ignatev , Mikhail Venchakov","doi":"10.1016/j.exmath.2025.125656","DOIUrl":"10.1016/j.exmath.2025.125656","url":null,"abstract":"<div><div>Let <span><math><mi>U</mi></math></span> be the unitriangular group over a finite field. We consider an interesting class of irreducible complex characters of <span><math><mi>U</mi></math></span>, so-called characters of depth 2. This is a next natural step after characters of maximal and submaximal dimension, whose description is already known. We explicitly describe the support of a character of depth 2 by a system of defining algebraic equations. After that, we calculate the value of such a character on an element from the support. The main technical tool used in the proofs is the Mackey little group method for semidirect products.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 3","pages":"Article 125656"},"PeriodicalIF":0.8,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349100","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Dualistic structures in information geometry","authors":"Leonard Todjihounde","doi":"10.1016/j.exmath.2025.125654","DOIUrl":"10.1016/j.exmath.2025.125654","url":null,"abstract":"<div><div>Important basics on dualistic structures on Riemannian manifolds are revisited and presented as a fundamental concept connecting information geometry, affine geometry and Hessian geometry. Since several statistical manifolds can be seen as warped product spaces, we conclude this survey by some results on warped products of dualistic structures.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 3","pages":"Article 125654"},"PeriodicalIF":0.8,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143386889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Statistically characterized subgroups related to some non-arithmetic sequence of integers","authors":"Pratulananda Das, Ayan Ghosh","doi":"10.1016/j.exmath.2025.125653","DOIUrl":"10.1016/j.exmath.2025.125653","url":null,"abstract":"<div><div>Very recently in Das and Ghosh (2024), characterized subgroups have been investigated for some special kind of non-arithmetic sequences where certain cardinality related questions were answered. As statistically characterized subgroups Dikranjan et al. (2020) have evolved as non-trivial generalization of characterized subgroups, it is natural to ask the same questions for these subgroups which we try to answer here. The entire investigation emphasizes that these statistically characterized subgroups are mostly larger in size, having cardinality <span><math><mi>c</mi></math></span>, and exhibit behavior that significantly differs from that of classical characterized subgroups. As a consequence, we are able to present solution of an open problem raised in Dikranjan et al. (2020).</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 3","pages":"Article 125653"},"PeriodicalIF":0.8,"publicationDate":"2025-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Quelques considérations galoisiennes relatives à l’extension des constantes d’un corps de fractions tordu","authors":"Bruno Deschamps","doi":"10.1016/j.exmath.2024.125645","DOIUrl":"10.1016/j.exmath.2024.125645","url":null,"abstract":"<div><div>In this article, we state several results relating to the arithmetic of a constants extension of a skew fractions field <span><math><mrow><mi>K</mi><mrow><mo>[</mo><mi>t</mi><mo>,</mo><mi>σ</mi><mo>,</mo><mi>δ</mi><mo>]</mo></mrow></mrow></math></span>. As an application, we show a non-commutative version of the Leptin–Waterhouse theorem: for any profinite group <span><math><mi>Γ</mi></math></span>, there exist a skew field <span><math><mi>K</mi></math></span> and an algebraic, outer and Galois extension <span><math><mrow><mi>L</mi><mo>/</mo><mi>K</mi></mrow></math></span> such that <span><math><mrow><mtext>Gal</mtext><mrow><mo>(</mo><mi>L</mi><mo>/</mo><mi>K</mi><mo>)</mo></mrow><mo>≃</mo><mi>Γ</mi></mrow></math></span>.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 3","pages":"Article 125645"},"PeriodicalIF":0.8,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349058","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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