{"title":"Galois groups of random polynomials over the rational function field","authors":"Alexei Entin","doi":"10.1112/jlms.70061","DOIUrl":"https://doi.org/10.1112/jlms.70061","url":null,"abstract":"<p>For a fixed prime power <span></span><math>\u0000 <semantics>\u0000 <mi>q</mi>\u0000 <annotation>$q$</annotation>\u0000 </semantics></math> and natural number <span></span><math>\u0000 <semantics>\u0000 <mi>d</mi>\u0000 <annotation>$d$</annotation>\u0000 </semantics></math>, we consider a random polynomial\u0000\u0000 </p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143118215","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Groups acting on veering pairs and Kleinian groups","authors":"Hyungryul Baik, Hongtaek Jung, KyeongRo Kim","doi":"10.1112/jlms.70052","DOIUrl":"https://doi.org/10.1112/jlms.70052","url":null,"abstract":"<p>We show that some subgroups of the orientation-preserving circle homeomorphism group with invariant veering pairs of laminations are hyperbolic 3-orbifold groups. On the way, we show that from a veering pair of laminations, one can construct a loom space (in the sense of Schleimer–Segerman) as a quotient. Our approach does not assume the existence of any 3-manifold to begin with, so this is a geometrization-type result, and supersedes some of the results regarding the relation among veering triangulations, pseudo-Anosov flows, taut foliations in the literature.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143118216","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Galen Dorpalen-Barry, Joshua Maglione, Christian Stump
{"title":"The Poincaré-extended \u0000 \u0000 \u0000 a\u0000 b\u0000 \u0000 $mathbf {a}mathbf {b}$\u0000 -index","authors":"Galen Dorpalen-Barry, Joshua Maglione, Christian Stump","doi":"10.1112/jlms.70054","DOIUrl":"https://doi.org/10.1112/jlms.70054","url":null,"abstract":"<p>Motivated by a conjecture concerning Igusa local zeta functions for intersection posets of hyperplane arrangements, we introduce and study the <i>Poincaré-extended</i> <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mi>b</mi>\u0000 </mrow>\u0000 <annotation>$mathbf {a}mathbf {b}$</annotation>\u0000 </semantics></math><i>-index</i>, which generalizes both the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mi>b</mi>\u0000 </mrow>\u0000 <annotation>$mathbf {a}mathbf {b}$</annotation>\u0000 </semantics></math>-index and the Poincaré polynomial. For posets admitting <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$R$</annotation>\u0000 </semantics></math>-labelings, we give a combinatorial description of the coefficients of the extended <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mi>b</mi>\u0000 </mrow>\u0000 <annotation>$mathbf {a}mathbf {b}$</annotation>\u0000 </semantics></math>-index, proving their nonnegativity. In the case of intersection posets of hyperplane arrangements, we prove the above conjecture of the second author and Voll as well as another conjecture of the second author and Kühne. We also define the pullback <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mi>b</mi>\u0000 </mrow>\u0000 <annotation>$mathbf {a}mathbf {b}$</annotation>\u0000 </semantics></math>-index, generalizing the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>c</mi>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation>$mathbf {c}mathbf {d}$</annotation>\u0000 </semantics></math>-index of face posets for oriented matroids. Our results recover, generalize, and unify results from Billera–Ehrenborg–Readdy, Bergeron–Mykytiuk–Sottile–van Willigenburg, Saliola–Thomas, and Ehrenborg. This connection allows us to translate our results into the language of quasisymmetric functions, and — in the special case of symmetric functions — pose a conjecture about Schur positivity. This conjecture was strengthened and proved by Ricky Liu, and the proof appears as an appendix.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70054","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142868871","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}