Indagationes Mathematicae-New Series最新文献

{"title":"A characterization of differential operators in the ring of complex polynomials","authors":"Włodzimierz Fechner ,&nbsp;Eszter Gselmann","doi":"10.1016/j.indag.2024.07.011","DOIUrl":"10.1016/j.indag.2024.07.011","url":null,"abstract":"<div><div>The paper aims to provide a full characterization of all operators <span><math><mrow><mi>T</mi><mo>:</mo><mi>P</mi><mrow><mo>(</mo><mi>ℂ</mi><mo>)</mo></mrow><mo>→</mo><mi>P</mi><mrow><mo>(</mo><mi>ℂ</mi><mo>)</mo></mrow></mrow></math></span> acting on the space of all complex polynomials that satisfy the Leibniz rule <span><span><span><math><mrow><mi>T</mi><mrow><mo>(</mo><mi>f</mi><mi>⋅</mi><mi>g</mi><mo>)</mo></mrow><mo>=</mo><mi>T</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mi>⋅</mi><mi>g</mi><mo>+</mo><mi>f</mi><mi>⋅</mi><mi>T</mi><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow></mrow></math></span></span></span>for all <span><math><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>∈</mo><mi>P</mi><mrow><mo>(</mo><mi>ℂ</mi><mo>)</mo></mrow></mrow></math></span>. We do not assume the linearity of <span><math><mi>T</mi></math></span>. As we will see, contrary to the well-known theorems for function spaces there are many other solutions here, not only differential operators. From our main result, we also derive two corollaries, showing that in some special cases operators that satisfy the Leibniz rule have some particular form.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 6","pages":"Pages 1294-1305"},"PeriodicalIF":0.5,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142571884","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":"Finite odometer factors of rank one group actions","authors":"","doi":"10.1016/j.indag.2024.04.012","DOIUrl":"10.1016/j.indag.2024.04.012","url":null,"abstract":"<div><div><span>In this paper, we give explicit conditions characterizing the Følner rank one group actions that factor onto a finite odometer; those that factor onto an arbitrary, but specified odometer, and those that factor onto an unspecified odometer. We also give explicit conditions describing the Følner rank one actions that are conjugate to a specific odometer, and those that are conjugate to some odometer. These conditions are based on cutting and stacking procedures used to generate the action, and generalize results given by Foreman, Gao, Hill, Silva and Weiss for rank one </span><span><math><mi>Z</mi></math></span>-actions.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 6","pages":"Pages 1105-1137"},"PeriodicalIF":0.5,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141032515","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":"A note on the inverse problem for finite differential Galois groups","authors":"","doi":"10.1016/j.indag.2024.06.005","DOIUrl":"10.1016/j.indag.2024.06.005","url":null,"abstract":"<div><div>In this paper we revisit the following inverse problem: given a curve invariant under an irreducible finite linear algebraic group, can we construct an ordinary linear differential equation whose Schwarz map parametrizes it? We present an algorithmic solution to this problem under the assumption that we are given the function field of the quotient curve. The result provides a generalization and an efficient implementation of the solution to the inverse problem exposed by van der Put et al. (2020). As an application, we show that there is no hypergeometric equation with projective differential Galois group isomorphic to <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>72</mn></mrow></msub></math></span>, thus completing Beukers and Heckman’s answer (Beukers and Heckman, 1989) to the question of which irreducible finite subgroup of <span><math><mrow><mi>P</mi><mi>S</mi><msub><mrow><mi>L</mi></mrow><mrow><mn>3</mn></mrow></msub><mrow><mo>(</mo><mi>ℂ</mi><mo>)</mo></mrow></mrow></math></span> are the projective monodromy of a hypergeometric equation.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 6","pages":"Pages 1259-1269"},"PeriodicalIF":0.5,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141569879","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":"An analogue of a conjecture of Rasmussen and Tamagawa for abelian varieties over function fields","authors":"Mentzelos Melistas","doi":"10.1016/j.indag.2024.06.007","DOIUrl":"10.1016/j.indag.2024.06.007","url":null,"abstract":"<div><div>Let <span><math><mi>L</mi></math></span> be a number field and let <span><math><mi>ℓ</mi></math></span> be a prime number. Rasmussen and Tamagawa conjectured, in a precise sense, that abelian varieties whose field of definition of the <span><math><mi>ℓ</mi></math></span>-power torsion is both a pro-<span><math><mi>ℓ</mi></math></span> extension of <span><math><mrow><mi>L</mi><mrow><mo>(</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> and unramified away from <span><math><mi>ℓ</mi></math></span> are quite rare. In this paper, we formulate an analogue of the Rasmussen–Tamagawa conjecture for non-isotrivial abelian varieties defined over function fields. We provide a proof of our analogue in the case of elliptic curves. In higher dimensions, when the base field is a subfield of the complex numbers, we show that our conjecture is a consequence of the uniform geometric torsion conjecture. Finally, using a theorem of Bakker and Tsimerman we also prove our conjecture unconditionally for abelian varieties with real multiplication.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 6","pages":"Pages 1270-1281"},"PeriodicalIF":0.5,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141690718","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":"The Generalized Riemann Hypothesis from zeros of a single L-function","authors":"William Banks","doi":"10.1016/j.indag.2024.07.009","DOIUrl":"10.1016/j.indag.2024.07.009","url":null,"abstract":"<div><div>For each primitive Dirichlet character <span><math><mi>χ</mi></math></span>, a hypothesis <span><math><mrow><msup><mrow><mi>GRH</mi></mrow><mrow><mi>†</mi></mrow></msup><mrow><mo>[</mo><mi>χ</mi><mo>]</mo></mrow></mrow></math></span> is formulated in terms of zeros of the associated <span><math><mi>L</mi></math></span>-function <span><math><mrow><mi>L</mi><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>χ</mi><mo>)</mo></mrow></mrow></math></span>. It is shown that for any such character, <span><math><mrow><msup><mrow><mi>GRH</mi></mrow><mrow><mi>†</mi></mrow></msup><mrow><mo>[</mo><mi>χ</mi><mo>]</mo></mrow></mrow></math></span> is equivalent to the Generalized Riemann Hypothesis.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 6","pages":"Pages 1282-1293"},"PeriodicalIF":0.5,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141845529","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":"Non-Hausdorff germinal groupoids for actions of countable groups","authors":"","doi":"10.1016/j.indag.2024.05.005","DOIUrl":"10.1016/j.indag.2024.05.005","url":null,"abstract":"<div><div>We study conditions under which the germinal groupoid associated to a minimal equicontinuous action of a countable group on a Cantor set has non-Hausdorff topology. We develop a new criterion, which serves as an obstruction for the étale topology on the groupoid to be non-Hausdorff. We use this and other criteria to study the topology of germinal groupoids for a few classes of actions. In particular, we give examples of families of contracting and non-contracting self-similar groups, which are amenable and whose actions have associated germinal groupoids with non-Hausdorff topology.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 6","pages":"Pages 1149-1184"},"PeriodicalIF":0.5,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141569880","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":"Restriction theorems and root systems for symmetric superspaces","authors":"Shifra Reif ,&nbsp;Siddhartha Sahi ,&nbsp;Vera Serganova","doi":"10.1016/j.indag.2024.09.006","DOIUrl":"10.1016/j.indag.2024.09.006","url":null,"abstract":"<div><div>In this paper we consider those involutions <span><math><mi>θ</mi></math></span> of a finite-dimensional Kac–Moody Lie superalgebra <span><math><mi>g</mi></math></span>, with associated decomposition <span><math><mrow><mi>g</mi><mo>=</mo><mi>k</mi><mo>⊕</mo><mi>p</mi></mrow></math></span>, for which a Cartan subspace <span><math><mi>a</mi></math></span> in <span><math><msub><mrow><mi>p</mi></mrow><mrow><mover><mrow><mn>0</mn></mrow><mrow><mo>̄</mo></mrow></mover></mrow></msub></math></span> is self-centralizing in <span><math><mi>p</mi></math></span>. For such <span><math><mi>θ</mi></math></span> the restriction map <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>θ</mi></mrow></msub></math></span> from <span><math><mi>p</mi></math></span> to <span><math><mi>a</mi></math></span> is injective on the algebra <span><math><mrow><mi>P</mi><msup><mrow><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow></mrow><mrow><mi>k</mi></mrow></msup></mrow></math></span> of <span><math><mi>k</mi></math></span>-invariant polynomials on <span><math><mi>p</mi></math></span>. There are five infinite families and five exceptional cases of such involutions, and for each case we explicitly determine the structure of <span><math><mrow><mi>P</mi><msup><mrow><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow></mrow><mrow><mi>k</mi></mrow></msup></mrow></math></span> by giving a complete set of generators for the image of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>θ</mi></mrow></msub></math></span>. We also determine precisely when the restriction map <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>θ</mi></mrow></msub></math></span> from <span><math><mrow><mi>P</mi><msup><mrow><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow></mrow><mrow><mi>g</mi></mrow></msup></mrow></math></span> to <span><math><mrow><mi>P</mi><msup><mrow><mrow><mo>(</mo><mi>p</mi><mo>)</mo></mrow></mrow><mrow><mi>k</mi></mrow></msup></mrow></math></span> is surjective. Finally we introduce the notion of a generalized restricted root system, and show that in the present setting the <span><math><mi>a</mi></math></span>-roots <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow></mrow></math></span> always form such a system.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 2","pages":"Pages 609-630"},"PeriodicalIF":0.5,"publicationDate":"2024-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143463425","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":"Non-symmetric Jacobi polynomials of type BC1 as vector-valued polynomials, Part 1: Spherical functions","authors":"M. van Horssen,&nbsp;M. van Pruijssen","doi":"10.1016/j.indag.2024.09.003","DOIUrl":"10.1016/j.indag.2024.09.003","url":null,"abstract":"<div><div>We study non-symmetric Jacobi polynomials of type <span><math><mrow><mi>B</mi><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span> by means of vector-valued and matrix-valued orthogonal polynomials. The interpretation as matrix-valued orthogonal polynomials yields a new expression of the non-symmetric Jacobi polynomials of type <span><math><mrow><mi>B</mi><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span> in terms of the symmetric Jacobi polynomials of type <span><math><mrow><mi>B</mi><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span>. In this interpretation, the Cherednik operator, that has the non-symmetric Jacobi polynomials as eigenfunctions, corresponds to two shift operators for the symmetric Jacobi polynomials of type <span><math><mrow><mi>B</mi><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span>.</div><div>We show that the non-symmetric Jacobi polynomials of type <span><math><mrow><mi>B</mi><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span> with so-called geometric root multiplicities, interpreted as vector-valued polynomials, can be identified with spherical functions on the sphere <span><math><mrow><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>=</mo><mi>Spin</mi><mrow><mo>(</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mo>/</mo><mi>Spin</mi><mrow><mo>(</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> associated with the fundamental spin-representation of <span><math><mrow><mi>Spin</mi><mrow><mo>(</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>. The Cherednik operator corresponds to the Dirac operator for the spinors on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> in this interpretation.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 2","pages":"Pages 593-608"},"PeriodicalIF":0.5,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143463561","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":"Critical curves of rotations","authors":"","doi":"10.1016/j.indag.2024.02.001","DOIUrl":"10.1016/j.indag.2024.02.001","url":null,"abstract":"<div><p>In rotations with a binary symbolic dynamics, a critical curve is the locus of parameters for which the boundaries of the partition that defines the symbolic dynamics are connected via a prescribed number of iterations and symbolic itinerary. We study the arithmetical and geometrical properties of these curves in parameter space.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 5","pages":"Pages 989-1008"},"PeriodicalIF":0.5,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357724000089/pdfft?md5=2f9d5a6610f18fbac9a8c979bf5335b4&pid=1-s2.0-S0019357724000089-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139679261","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":"Semicocycle discontinuities for substitutions and reverse-reading automata","authors":"","doi":"10.1016/j.indag.2023.05.003","DOIUrl":"10.1016/j.indag.2023.05.003","url":null,"abstract":"<div><p>In this article we define the semigroup associated to a primitive substitution. We use it to construct a minimal automaton which generates a substitution sequence <span><math><mi>u</mi></math></span> in reverse reading. We show, in the case where the substitution has a coincidence, that this automaton completely describes the <em>semicocycle discontinuities</em> of <span><math><mi>u</mi></math></span>.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 5","pages":"Pages 796-812"},"PeriodicalIF":0.5,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357723000496/pdfft?md5=0a7daca4bb4987075ce1de3e272a8290&pid=1-s2.0-S0019357723000496-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46621925","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}