Pub Date : 2025-07-01Epub Date: 2025-02-24DOI: 10.1016/j.exmath.2025.125660
Maria Stella Adamo , Karl-Hermann Neeb , Jonas Schober
We present a novel perspective on reflection positivity on the strip by systematically developing the analogies with the unit disc and the upper half plane in the complex plane. These domains correspond to the three conjugacy classes of one-parameter groups in the Möbius group (elliptic for the disc, parabolic for the upper half plane and hyperbolic for the strip). In all cases, reflection positive functions correspond to positive functionals on for a suitable involution. For the strip, reflection positivity naturally connects with Kubo–Martin–Schwinger (KMS) conditions on the real line and further to standard pairs, as they appear in Algebraic Quantum Field Theory. We also exhibit a curious connection between Hilbert spaces on the strip and the upper half plane, based on a periodization process.
{"title":"Reflection positivity and its relation to disc, half plane and the strip","authors":"Maria Stella Adamo , Karl-Hermann Neeb , Jonas Schober","doi":"10.1016/j.exmath.2025.125660","DOIUrl":"10.1016/j.exmath.2025.125660","url":null,"abstract":"<div><div>We present a novel perspective on reflection positivity on the strip by systematically developing the analogies with the unit disc and the upper half plane in the complex plane. These domains correspond to the three conjugacy classes of one-parameter groups in the Möbius group (elliptic for the disc, parabolic for the upper half plane and hyperbolic for the strip). In all cases, reflection positive functions correspond to positive functionals on <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span> for a suitable involution. For the strip, reflection positivity naturally connects with Kubo–Martin–Schwinger (KMS) conditions on the real line and further to standard pairs, as they appear in Algebraic Quantum Field Theory. We also exhibit a curious connection between Hilbert spaces on the strip and the upper half plane, based on a periodization process.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 4","pages":"Article 125660"},"PeriodicalIF":0.8,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143579109","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}
Pub Date : 2025-07-01Epub Date: 2025-04-03DOI: 10.1016/j.exmath.2025.125686
Jingsheng Wang, Pengtong Li
In this paper, we extend the redundancy theorem of J. Gabardo and D. Han for Gabor type unitary systems indexed by full-rank lattices in Euclidean spaces to the setting of locally compact abelian (LCA) groups. Let be an LCA group, let be a uniform lattice in , let be an automorphism of , and let be an automorphism of . We show that the redundancy of a Gabor type unitary system indexed by equals the reciprocal of the density of this index set. As an application, we give a new proof of the famous time-frequency density theorem in Gabor analysis.
{"title":"The redundancy of Gabor type unitary systems on locally compact abelian groups","authors":"Jingsheng Wang, Pengtong Li","doi":"10.1016/j.exmath.2025.125686","DOIUrl":"10.1016/j.exmath.2025.125686","url":null,"abstract":"<div><div>In this paper, we extend the redundancy theorem of J. Gabardo and D. Han for Gabor type unitary systems indexed by full-rank lattices in Euclidean spaces to the setting of locally compact abelian (LCA) groups. Let <span><math><mi>G</mi></math></span> be an LCA group, let <span><math><mi>Λ</mi></math></span> be a uniform lattice in <span><math><mi>G</mi></math></span>, let <span><math><mi>α</mi></math></span> be an automorphism of <span><math><mi>G</mi></math></span>, and let <span><math><mi>β</mi></math></span> be an automorphism of <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>̂</mo></mrow></mover></math></span>. We show that the redundancy of a Gabor type unitary system indexed by <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>Λ</mi><mo>)</mo></mrow><mo>×</mo><mi>β</mi><mrow><mo>(</mo><msup><mrow><mi>Λ</mi></mrow><mrow><mo>⊥</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> equals the reciprocal of the density of this index set. As an application, we give a new proof of the famous time-frequency density theorem in Gabor analysis.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 4","pages":"Article 125686"},"PeriodicalIF":0.8,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808262","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}
Pub Date : 2025-07-01Epub Date: 2025-04-03DOI: 10.1016/j.exmath.2025.125688
Ashleigh Ratcliffe, Bogdan Grechuk
Generalized Fermat equation (GFE) is the equation of the form , where are positive integers. If , GFE is known to have at most finitely many primitive integer solutions . A large body of the literature is devoted to finding such solutions explicitly for various six-tuples , as well as for infinite families of such six-tuples. This paper surveys the families of parameters for which GFE has been solved. Although the proofs are not discussed here, collecting these references in one place will make it easier for the readers to find the relevant proof techniques in the original papers. Also, this survey will help the readers to avoid duplicate work by solving the already solved cases.
{"title":"Generalized Fermat equation: A survey of solved cases","authors":"Ashleigh Ratcliffe, Bogdan Grechuk","doi":"10.1016/j.exmath.2025.125688","DOIUrl":"10.1016/j.exmath.2025.125688","url":null,"abstract":"<div><div>Generalized Fermat equation (GFE) is the equation of the form <span><math><mrow><mi>a</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>b</mi><msup><mrow><mi>y</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>=</mo><mi>c</mi><msup><mrow><mi>z</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></math></span>, where <span><math><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>r</mi></mrow></math></span> are positive integers. If <span><math><mrow><mn>1</mn><mo>/</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>/</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>/</mo><mi>r</mi><mo><</mo><mn>1</mn></mrow></math></span>, GFE is known to have at most finitely many primitive integer solutions <span><math><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow></math></span>. A large body of the literature is devoted to finding such solutions explicitly for various six-tuples <span><math><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></math></span>, as well as for infinite families of such six-tuples. This paper surveys the families of parameters for which GFE has been solved. Although the proofs are not discussed here, collecting these references in one place will make it easier for the readers to find the relevant proof techniques in the original papers. Also, this survey will help the readers to avoid duplicate work by solving the already solved cases.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 4","pages":"Article 125688"},"PeriodicalIF":0.8,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143839742","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}
Pub Date : 2025-07-01Epub Date: 2025-03-27DOI: 10.1016/j.exmath.2025.125679
Antigona Pajaziti , Mohammad Sadek
Let be an elliptic curve defined over and denote the reduction of modulo a prime of good reduction for . The divisibility of by an integer for a set of primes of density 1 is determined by the torsion subgroups of elliptic curves that are -isogenous to . In this work, we give explicit families of elliptic curves over together with integers such that the congruence class of modulo can be computed explicitly. In addition, we can estimate the density of primes for which each congruence class occurs. These include elliptic curves over whose torsion grows over a quadratic field where is determined by the -torsion subgroups in the -isogeny class of . We also exhibit elliptic curves over for which the orders of the reductions of every smooth fiber modulo primes of positive density strictly less than 1 are divisible by given small integers.
{"title":"Divisibility of orders of reductions of elliptic curves","authors":"Antigona Pajaziti , Mohammad Sadek","doi":"10.1016/j.exmath.2025.125679","DOIUrl":"10.1016/j.exmath.2025.125679","url":null,"abstract":"<div><div>Let <span><math><mi>E</mi></math></span> be an elliptic curve defined over <span><math><mi>Q</mi></math></span> and <span><math><msub><mrow><mover><mrow><mi>E</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>p</mi></mrow></msub></math></span> denote the reduction of <span><math><mi>E</mi></math></span> modulo a prime <span><math><mi>p</mi></math></span> of good reduction for <span><math><mi>E</mi></math></span>. The divisibility of <span><math><mrow><mo>|</mo><msub><mrow><mover><mrow><mi>E</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></mrow><mo>|</mo></mrow></math></span> by an integer <span><math><mrow><mi>m</mi><mo>≥</mo><mn>2</mn></mrow></math></span> for a set of primes <span><math><mi>p</mi></math></span> of density 1 is determined by the torsion subgroups of elliptic curves that are <span><math><mi>Q</mi></math></span>-isogenous to <span><math><mi>E</mi></math></span>. In this work, we give explicit families of elliptic curves <span><math><mi>E</mi></math></span> over <span><math><mi>Q</mi></math></span> together with integers <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>E</mi></mrow></msub></math></span> such that the congruence class of <span><math><mrow><mo>|</mo><msub><mrow><mover><mrow><mi>E</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></mrow><mo>|</mo></mrow></math></span> modulo <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>E</mi></mrow></msub></math></span> can be computed explicitly. In addition, we can estimate the density of primes <span><math><mi>p</mi></math></span> for which each congruence class occurs. These include elliptic curves over <span><math><mi>Q</mi></math></span> whose torsion grows over a quadratic field <span><math><mi>K</mi></math></span> where <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>E</mi></mrow></msub></math></span> is determined by the <span><math><mi>K</mi></math></span>-torsion subgroups in the <span><math><mi>Q</mi></math></span>-isogeny class of <span><math><mi>E</mi></math></span>. We also exhibit elliptic curves over <span><math><mrow><mi>Q</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> for which the orders of the reductions of every smooth fiber modulo primes of positive density strictly less than 1 are divisible by given small integers.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 4","pages":"Article 125679"},"PeriodicalIF":0.8,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143747610","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}
Pub Date : 2025-07-01Epub Date: 2025-04-01DOI: 10.1016/j.exmath.2025.125680
Qiyao Bao , Deguang Han , Rui Liu , Jie Shen
Nonlinear framings naturally appear in many applications where nonlinear procedures are necessary. This paper examines two basic issues involving the linearization of Lipschitz framings. We first prove that every Lipschitz framing induces a linear framing which shares the same synthesis operator, and consequently every Banach space admitting a Lipschitz framing has the bounded approximation property. Secondly, we examine the projection-valued dilations of Lipschitz operator-valued measures on Banach spaces. We prove that every Lipschitz operator-valued measure can induce an operator-valued measure by linearization, and every -valued measure has a projection-valued measure dilation by establishing a nonlinear version of minimal dilation theory. As examples, we discuss a concrete construction of the minimal dilation for the special case when the measure space is , and how nonlinear sampling naturally induces a Lipschitz framing.
{"title":"Linearization of Lipschitz framings for Banach spaces","authors":"Qiyao Bao , Deguang Han , Rui Liu , Jie Shen","doi":"10.1016/j.exmath.2025.125680","DOIUrl":"10.1016/j.exmath.2025.125680","url":null,"abstract":"<div><div>Nonlinear framings naturally appear in many applications where nonlinear procedures are necessary. This paper examines two basic issues involving the linearization of Lipschitz framings. We first prove that every Lipschitz framing induces a linear framing which shares the same synthesis operator, and consequently every Banach space admitting a Lipschitz framing has the bounded approximation property. Secondly, we examine the projection-valued dilations of Lipschitz operator-valued measures on Banach spaces. We prove that every Lipschitz operator-valued measure can induce an operator-valued measure by linearization, and every <span><math><mrow><mi>Lip</mi><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></mrow></mrow></math></span>-valued measure has a projection-valued measure dilation by establishing a nonlinear version of minimal dilation theory. As examples, we discuss a concrete construction of the minimal dilation for the special case when the measure space is <span><math><mrow><mo>(</mo><mi>N</mi><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo></mrow></math></span>, and how nonlinear sampling naturally induces a Lipschitz framing.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 4","pages":"Article 125680"},"PeriodicalIF":0.8,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143768160","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}
Pub Date : 2025-07-01Epub Date: 2025-04-02DOI: 10.1016/j.exmath.2025.125685
Emilie Mai Elkiær
We show via an application of techniques from complex interpolation theory how the -pseudofunction algebras of a locally compact group can be understood as sitting between and . Motivated by this, we collect and review various characterizations of group amenability connected to the -pseudofunction algebra of Herz and generalize these to the symmetrized setting. Along the way, we describe the Banach space dual of the symmetrized pseudofunction algebras on associated with representations on reflexive Banach spaces.
我们通过应用复插值理论的技术，说明局部紧凑群 G 的 Lp 伪函数代数如何被理解为介于 L1(G) 和 C∗(G) 之间。受此启发，我们收集并回顾了与赫兹的 p 伪函数代数相关的各种群可亲性特征，并将这些特征推广到对称设置中。同时，我们还描述了与反身巴拿赫空间上的表征相关的 G 上对称伪函数代数的巴拿赫空间对偶。
{"title":"Symmetrized pseudofunction algebras from Lp-representations and amenability of locally compact groups","authors":"Emilie Mai Elkiær","doi":"10.1016/j.exmath.2025.125685","DOIUrl":"10.1016/j.exmath.2025.125685","url":null,"abstract":"<div><div>We show via an application of techniques from complex interpolation theory how the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-pseudofunction algebras of a locally compact group <span><math><mi>G</mi></math></span> can be understood as sitting between <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msup><mrow><mi>C</mi></mrow><mrow><mo>∗</mo></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Motivated by this, we collect and review various characterizations of group amenability connected to the <span><math><mi>p</mi></math></span>-pseudofunction algebra of Herz and generalize these to the symmetrized setting. Along the way, we describe the Banach space dual of the symmetrized pseudofunction algebras on <span><math><mi>G</mi></math></span> associated with representations on reflexive Banach spaces.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 4","pages":"Article 125685"},"PeriodicalIF":0.8,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143820722","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}
Pub Date : 2025-05-01Epub Date: 2025-01-25DOI: 10.1016/j.exmath.2025.125657
Konrad Schmüdgen
Suppose that is a finitely generated commutative unital real algebra and is a closed subset of the set of characters of . We study the following problem: When is each linear functional an integral with respect to some signed Radon measure on supported by the set ? A complete characterization of these sets and algebras by necessary and sufficient conditions is given. The result is applied to the polynomial algebra and subsets of .
{"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":"2025-05-01","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}
Pub Date : 2025-05-01Epub Date: 2025-01-10DOI: 10.1016/j.exmath.2024.125645
Bruno Deschamps
In this article, we state several results relating to the arithmetic of a constants extension of a skew fractions field . As an application, we show a non-commutative version of the Leptin–Waterhouse theorem: for any profinite group , there exist a skew field and an algebraic, outer and Galois extension such that .
{"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":"2025-05-01","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}
Pub Date : 2025-05-01Epub Date: 2025-02-04DOI: 10.1016/j.exmath.2025.125659
Luc Guyot
Let be the ring of univariate polynomials over and denote by its stable rank in the sense of Bass. Grunewald, Mennicke and Vaserstein proved that As the inequality 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 is not stable.
{"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":"2025-05-01","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}
Pub Date : 2025-05-01Epub Date: 2025-01-14DOI: 10.1016/j.exmath.2025.125653
Pratulananda Das, Ayan Ghosh
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 , 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).
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