Indagationes Mathematicae-New Series最新文献

{"title":"Basic quasi-reductive root data and supergroups","authors":"Rita Fioresi,&nbsp;Bin Shu","doi":"10.1016/j.indag.2026.03.005","DOIUrl":"10.1016/j.indag.2026.03.005","url":null,"abstract":"<div><div>We investigate pairs <span><math><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>Y</mi><mo>)</mo></mrow></math></span>, where <span><math><mi>G</mi></math></span> is a reductive algebraic group and <span><math><mi>Y</mi></math></span> a purely-odd <span><math><mi>G</mi></math></span>-superscheme, asking when a pair corresponds to a quasi-reductive algebraic supergroup <span><math><mi>G</mi></math></span>, that is, <span><math><msub><mrow><mi>G</mi></mrow><mrow><mtext>ev</mtext></mrow></msub></math></span> is isomorphic to <span><math><mi>G</mi></math></span>, and the quotient <span><math><mrow><mi>G</mi><mo>/</mo><msub><mrow><mi>G</mi></mrow><mrow><mtext>ev</mtext></mrow></msub></mrow></math></span> is <span><math><mi>G</mi></math></span>-equivariantly isomorphic to <span><math><mi>Y</mi></math></span>. We prove that, if <span><math><mi>Y</mi></math></span> satisfies certain conditions (basic quasi-reductive root data), then the question has a positive answer given by an existence and uniqueness theorem. The corresponding supergroups are said to be basic quasi-reductive, which can be classified, up to isogeny. We then decide the structure of connected quasi-reductive algebraic supergroups provided that: (i) the root system does not contain 0; (ii) <span><math><mrow><mi>g</mi><mo>≔</mo><mtext>Lie</mtext><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> admits a nondegenerate even super-symmetric bilinear form. (iii) all odd reflections are invertible. Remarkably, those supergroups are exactly basic quasi-reductive supergroups of monodromy type.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 5","pages":"Pages 1917-1933"},"PeriodicalIF":0.8,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148916870","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":"The magic and mystery of component tableaux","authors":"Yasmine Fittouhi,&nbsp;Anthony Joseph","doi":"10.1016/j.indag.2025.12.003","DOIUrl":"10.1016/j.indag.2025.12.003","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; be a simple algebraic group over the complex field &lt;span&gt;&lt;math&gt;&lt;mi&gt;ℂ&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; a fixed Borel subgroup, &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; a parabolic subgroup containing &lt;span&gt;&lt;math&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; its derived group and &lt;span&gt;&lt;math&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; the Lie algebra of its nilradical.&lt;/div&gt;&lt;div&gt;The nilfibre &lt;span&gt;&lt;math&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; for this action is the zero locus of the augmentation &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℐ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; of the semi-invariant algebra &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;ℐ&lt;/mi&gt;&lt;mo&gt;≔&lt;/mo&gt;&lt;mi&gt;ℂ&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;Although the Lasker–Noether theorem allows one to define the components of an algebraic variety, their determination is notoriously difficult, especially for &lt;span&gt;&lt;math&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; as it is homogeneous. Even for &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; practically nothing was known previously. The only result of comparable, but lesser complexity, is for &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;≔&lt;/mo&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo&gt;∩&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, with &lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; a nilpotent &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; orbit and &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; the set of strictly upper triangular matrices. Then &lt;span&gt;&lt;math&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is equi-dimensional, with components, known as orbital varieties, parameterized by standard tableaux whose shape is dictated by &lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. This result is made relatively easy through Bruhat decomposition.&lt;/div&gt;&lt;div&gt;Here the components of &lt;span&gt;&lt;math&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; are studied for &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. They increase exponentially in &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; with no a priori discernible pattern. Nor is it possible to use Bruhat decomposition.&lt;/div&gt;&lt;div&gt;The composition of &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; defined by the Levi block sizes in &lt;span&gt;&lt;math&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; defines a standard tableau &lt;span&gt;&lt;math&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. For each choice of numerical data &lt;span&gt;&lt;math&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, a semi-standard tableau &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, is constructed from &lt;span&gt;&lt;math&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. A &lt;em&gt;delicate and tightly interlocking&lt;/em&gt; analysis constructs a set of excluded root vectors from &lt;span&gt;&lt;math&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; such that the complementary space &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 5","pages":"Pages 1635-1696"},"PeriodicalIF":0.8,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148916964","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":"Representing integers as sums of mixed powers of primes","authors":"Geovane Matheus Lemes Andrade,&nbsp;Hemar Godinho","doi":"10.1016/j.indag.2026.01.001","DOIUrl":"10.1016/j.indag.2026.01.001","url":null,"abstract":"<div><div>We establish two new Waring–Goldbach type representations: every sufficiently large odd integer <span><math><mi>n</mi></math></span> can be expressed as <span><span><span><math><mrow><mi>n</mi><mo>=</mo><msubsup><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>3</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>p</mi></mrow><mrow><mn>4</mn></mrow><mrow><mn>3</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>p</mi></mrow><mrow><mn>5</mn></mrow><mrow><mn>5</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>p</mi></mrow><mrow><mn>6</mn></mrow><mrow><mn>6</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>p</mi></mrow><mrow><mn>7</mn></mrow><mrow><mi>c</mi></mrow></msubsup><mo>,</mo></mrow></math></span></span></span>where each <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is prime and <span><math><mrow><mi>c</mi><mo>∈</mo><mrow><mo>{</mo><mn>6,7</mn><mo>}</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 5","pages":"Pages 1725-1732"},"PeriodicalIF":0.8,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148916966","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":"On the Markoff spectrum on the Hecke group of index six","authors":"Byungchul Cha,&nbsp;Dong Han Kim,&nbsp;Deokwon Sim","doi":"10.1016/j.indag.2026.01.004","DOIUrl":"10.1016/j.indag.2026.01.004","url":null,"abstract":"<div><div>The discrete part of the Markoff spectrum on the Hecke group of index 6 was determined by A. Schmidt. In this paper, we study its Markoff and Lagrange spectra after the smallest accumulation point <span><math><mrow><mn>4</mn><mo>/</mo><msqrt><mrow><mn>3</mn></mrow></msqrt></mrow></math></span>. We show that both the Markoff and Lagrange spectra below <span><math><mrow><mn>4</mn><mo>/</mo><msqrt><mrow><mn>3</mn></mrow></msqrt><mo>+</mo><mi>ϵ</mi></mrow></math></span> have positive Hausdorff dimension for any positive <span><math><mi>ϵ</mi></math></span>. We also find maximal gaps and an isolated point in the spectra.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 5","pages":"Pages 1776-1796"},"PeriodicalIF":0.8,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148916969","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":"Dynamical characterization of central sets in adequate partial semigroups","authors":"Pintu Debnath,&nbsp;Sayan Goswami,&nbsp;Sourav Kanti Patra","doi":"10.1016/j.indag.2026.02.002","DOIUrl":"10.1016/j.indag.2026.02.002","url":null,"abstract":"<div><div>Using tools from topological dynamics, H. Furstenberg introduced the notion of <em>central sets</em> and established the celebrated Central Sets Theorem, which simultaneously generalizes both van der Waerden’s theorem and Hindman’s theorem. Later, V. Bergelson and N. Hindman provided an equivalent description of central sets in <span><math><mi>N</mi></math></span> using the algebraic structure of the Stone–Čech compactification of discrete semigroups, and the full generality was subsequently obtained by H. Shi and H. Yang. Through ultrafilter techniques, J. McLeod further extended the concept of central sets to commutative adequate partial semigroups, with analogous definitions available for the noncommutative setting.</div><div>In this article, we introduce a framework of topological dynamics for actions of partial semigroups and, within this framework, we obtain a dynamical characterization of central sets in partial semigroups that is equivalent to the ultrafilter-based definitions.1<span><span>¹</span></span></div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 5","pages":"Pages 1834-1848"},"PeriodicalIF":0.8,"publicationDate":"2026-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148916867","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":"Weil polynomials of small degree","authors":"Stefano Marseglia","doi":"10.1016/j.indag.2025.07.008","DOIUrl":"10.1016/j.indag.2025.07.008","url":null,"abstract":"<div><div>Honda and Tate showed that the isogeny classes of abelian varieties of dimension <span><math><mi>g</mi></math></span> over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> are classified in terms of <span><math><mi>q</mi></math></span>-Weil polynomials of degree <span><math><mrow><mn>2</mn><mi>g</mi></mrow></math></span>, that is, monic integer polynomials whose set of complex roots consists of <span><math><mi>g</mi></math></span> conjugate pairs of absolute value <span><math><msqrt><mrow><mi>q</mi></mrow></msqrt></math></span>. There are descriptions of the space of such polynomials for <span><math><mrow><mi>g</mi><mo>≤</mo><mn>5</mn></mrow></math></span>, but for <span><math><mrow><mi>g</mi><mo>=</mo><mn>3</mn></mrow></math></span>, 4 and 5, these results contain mistakes. We correct these statements. Our proofs build on a criterion that determines when a real polynomial has only real roots in terms of the non-necessarily distinct roots of its first derivative.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 4","pages":"Pages 1093-1116"},"PeriodicalIF":0.8,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148546349","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":"Thom polynomials. A primer","authors":"R. Rimányi","doi":"10.1016/j.indag.2025.09.005","DOIUrl":"10.1016/j.indag.2025.09.005","url":null,"abstract":"<div><div>The Thom polynomial of a singularity <span><math><mi>η</mi></math></span> expresses the cohomology class of the <span><math><mi>η</mi></math></span>-singularity locus of a map in terms of the map’s simple invariants. In this survey, we provide an example-based summary of known results on various Thom polynomial concepts and explore the frontiers of this rich and evolving subject.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 4","pages":"Pages 1280-1305"},"PeriodicalIF":0.8,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148546656","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":"Spectral properties of the Gramian of finite ultrametric spaces","authors":"Gavin Robertson","doi":"10.1016/j.indag.2025.07.007","DOIUrl":"10.1016/j.indag.2025.07.007","url":null,"abstract":"&lt;div&gt;&lt;div&gt;The concept of &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-negative type is such that a metric space &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; has &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-negative type if and only if &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; embeds isometrically into a Hilbert space. If &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; then the &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-negative type of &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is intimately related to the Gramian matrix &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. In particular, &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; has strict &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-negative type if and only if &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is strictly positive semidefinite. As such, a natural measure of the degree of strictness of &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-negative type that &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; possesses is the minimum eigenvalue of the Gramian &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 4","pages":"Pages 1084-1092"},"PeriodicalIF":0.8,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148546651","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":"Trees and spectra of Heyting algebras","authors":"Damiano Fornasiere,&nbsp;Tommaso Moraschini","doi":"10.1016/j.indag.2025.08.002","DOIUrl":"10.1016/j.indag.2025.08.002","url":null,"abstract":"<div><div>A poset is <em>Esakia representable</em> when it is isomorphic to the prime spectrum of a Heyting algebra. Notably, every Esakia representable poset is also the spectrum of a commutative ring with unit. The problem of describing the Esakia representable posets was raised in 1985 and remains open to this day. We recall that a <em>forest</em> is a disjoint union of trees and that a <em>root system</em> is the order dual of a forest. It is shown that a root system is Esakia representable if and only if it satisfies a simple order theoretic condition, known as “having enough gaps”, and each of its nonempty chains has an infimum. This strengthens Lewis’s characterization of the root systems which are spectra of commutative rings with unit. While a similar characterization of arbitrary Esakia representable forests seems currently out of reach, we show that a <em>well-ordered</em> forest is Esakia representable if and only if it has enough gaps and each of its nonempty chains has a supremum.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 4","pages":"Pages 1158-1191"},"PeriodicalIF":0.8,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148539554","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":"A Hasse principle for GL2(Fp) and Bloch’s exact sequence for elliptic curves over number fields","authors":"Toshiro Hiranouchi","doi":"10.1016/j.indag.2025.10.005","DOIUrl":"10.1016/j.indag.2025.10.005","url":null,"abstract":"<div><div>We investigate the higher Chow groups, specifically <span><math><mrow><mi>S</mi><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> for elliptic curves <span><math><mi>E</mi></math></span> over number fields <span><math><mi>F</mi></math></span>. Focusing on the kernel <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> of the norm map <span><math><mrow><mi>S</mi><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mo>→</mo><msup><mrow><mi>F</mi></mrow><mrow><mo>×</mo></mrow></msup></mrow></math></span>, we analyze its mod <span><math><mi>p</mi></math></span> structure. We provide conditions, based on the mod <span><math><mi>p</mi></math></span> Galois representations associated to the <span><math><mi>p</mi></math></span>-torsion points <span><math><mrow><mi>E</mi><mrow><mo>[</mo><mi>p</mi><mo>]</mo></mrow></mrow></math></span>, under which the torsion subgroup of <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> is infinite.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 4","pages":"Pages 1476-1499"},"PeriodicalIF":0.8,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148546612","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}