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":"P-adic Poissonian pair correlations via the Monna map","authors":"Christian Weiß","doi":"10.1016/j.indag.2024.09.012","DOIUrl":"10.1016/j.indag.2024.09.012","url":null,"abstract":"<div><div>Although the existence of sequences in the p-adic integers with Poissonian pair correlations has already been shown, no explicit examples had been found so far. In this note we discuss how to transfer real sequences with Poissonian pair correlations to the p-adic setting by making use of the Monna map.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 3","pages":"Pages 912-919"},"PeriodicalIF":0.5,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143865108","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":"Pairing powers of pythagorean pairs","authors":"Lorenz Halbeisen ,&nbsp;Norbert Hungerbühler ,&nbsp;Arman Shamsi Zargar","doi":"10.1016/j.indag.2024.09.011","DOIUrl":"10.1016/j.indag.2024.09.011","url":null,"abstract":"<div><div>A pair <span><math><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></math></span> of positive integers is a <em>pythagorean pair</em> if <span><math><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span> is a square. A pythagorean pair <span><math><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></math></span> is called a <em>pythapotent pair of degree</em> <span><math><mi>h</mi></math></span> if there is another pythagorean pair <span><math><mrow><mo>(</mo><mi>k</mi><mo>,</mo><mi>l</mi><mo>)</mo></mrow></math></span>, which is not a multiple of <span><math><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></math></span>, such that <span><math><mrow><mo>(</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msup><mi>k</mi><mo>,</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>h</mi></mrow></msup><mi>l</mi><mo>)</mo></mrow></math></span> is a pythagorean pair. To each pythagorean pair <span><math><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></math></span> we assign an elliptic curve <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msup><mo>,</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>h</mi></mrow></msup></mrow></msub></math></span> for <span><math><mrow><mi>h</mi><mo>≥</mo><mn>3</mn></mrow></math></span> with torsion group isomorphic to <span><math><mrow><mi>Z</mi><mo>/</mo><mn>2</mn><mi>Z</mi><mo>×</mo><mi>Z</mi><mo>/</mo><mn>4</mn><mi>Z</mi></mrow></math></span> such that <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msup><mo>,</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>h</mi></mrow></msup></mrow></msub></math></span> has positive rank over <span><math><mi>Q</mi></math></span> if and only if <span><math><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></math></span> is a pythapotent pair of degree <span><math><mi>h</mi></math></span>. As a side result, we get that if <span><math><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></math></span> is a pythapotent pair of degree <span><math><mi>h</mi></math></span>, then there exist infinitely many pythagorean pairs <span><math><mrow><mo>(</mo><mi>k</mi><mo>,</mo><mi>l</mi><mo>)</mo></mrow></math></span>, not multiples of each other, such that <span><math><mrow><mo>(</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msup><mi>k</mi><mo>,</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>h</mi></mrow></msup><mi>l</mi><mo>)</mo></mrow></math></span> is a pythagorean pair. In particular, we show that any pythagorean pair is always a pythapotent pair of degree 3. In a previous work, pythapotent pairs of degrees 1 and 2 have been studied.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 3","pages":"Pages 903-911"},"PeriodicalIF":0.5,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143865107","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":"Waring’s problem for locally nilpotent groups: The case of discrete Heisenberg groups","authors":"Ya-Qing Hu","doi":"10.1016/j.indag.2024.09.010","DOIUrl":"10.1016/j.indag.2024.09.010","url":null,"abstract":"<div><div>Kamke (Kamke, 1921) solved an analog of Waring’s problem with <span><math><mi>n</mi></math></span>th power monomials replaced by integer-valued polynomials. Larsen and Nguyen (Larsen and Nguyen, 2019) explored the view of algebraic groups as a natural setting for Waring’s problem. This paper applies the theory of polynomial maps and polynomial sequences in locally nilpotent groups developed in a previous work (Hu, 2024) to solve an analog of Waring’s problem for the general discrete Heisenberg group <span><math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>Z</mi><mo>)</mo></mrow></mrow></math></span> for each integer <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math></span>.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 3","pages":"Pages 880-902"},"PeriodicalIF":0.5,"publicationDate":"2024-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143865106","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}