Journal of Pure and Applied Algebra最新文献

{"title":"Factorization of polynomials over the symmetrized tropical semiring and Descartes' rule of sign over ordered valued fields","authors":"Marianne Akian ,&nbsp;Stephane Gaubert ,&nbsp;Hanieh Tavakolipour","doi":"10.1016/j.jpaa.2025.108055","DOIUrl":"10.1016/j.jpaa.2025.108055","url":null,"abstract":"<div><div>The symmetrized tropical semiring is an extension of the tropical semifield, initially introduced to solve tropical linear systems using Cramer's rule. It is equivalent to the signed tropical hyperfield, which has been used in the study of tropicalizations of semialgebraic sets. Polynomials over the symmetrized tropical semiring, and their factorizations, were considered by Quadrat. Recently, Baker and Lorscheid introduced a notion of multiplicity for the roots of univariate polynomials over hyperfields. In the special case of the hyperfield of signs, they related multiplicities with Descartes' rule of signs for real polynomials. More recently, Gunn extended these multiplicity definitions and characterization to the setting of “whole idylls”. We investigate here the factorizations of univariate polynomial functions over symmetrized tropical semirings, and relate them to the multiplicities of roots over these semirings. We deduce Descartes' rule for “signs and valuations”, which applies to polynomials over a real closed field with a convex valuation and an arbitrary (divisible) value group. We show in particular that the inequality of Descartes' rule is tight when the value group is non-trivial. This extends a characterization of Gunn from the rank one case to arbitrary value groups, also answering the tightness question. Our results are obtained using the framework of semiring systems introduced by Rowen, together with model theory of valued fields.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 9","pages":"Article 108055"},"PeriodicalIF":0.7,"publicationDate":"2025-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144716114","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":"Kauffman bracket skein algebra of the 4-holed disk","authors":"Haimiao Chen","doi":"10.1016/j.jpaa.2025.108049","DOIUrl":"10.1016/j.jpaa.2025.108049","url":null,"abstract":"<div><div>We give a monomial basis for the Kauffman bracket skein algebra of the 4-holed disk, and find a presentation. This is based on an insight into the <span><math><mrow><mi>SL</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>C</mi><mo>)</mo></math></span>-character variety of the rank 4 free group.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 9","pages":"Article 108049"},"PeriodicalIF":0.7,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144687292","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":"Standard monomial theory modulo Frobenius in characteristic two","authors":"Laura Casabella ,&nbsp;Teresa Yu","doi":"10.1016/j.jpaa.2025.108051","DOIUrl":"10.1016/j.jpaa.2025.108051","url":null,"abstract":"<div><div>Over a field of characteristic two, we develop a theory of standard monomials for polynomial rings modulo a Frobenius power of the maximal ideal generated by all variables. As a result, we obtain a filtration by modular <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>-representations whose characters are given by particular truncated Schur polynomials, thus proving a conjecture by Gao–Raicu–VandeBogert in the characteristic two case.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 9","pages":"Article 108051"},"PeriodicalIF":0.7,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144687294","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":"Combinatorics of infinite rank module categories over finite dimensional sl3-modules in Lie-algebraic context","authors":"Volodymyr Mazorchuk ,&nbsp;Xiaoyu Zhu","doi":"10.1016/j.jpaa.2025.108054","DOIUrl":"10.1016/j.jpaa.2025.108054","url":null,"abstract":"<div><div>We determine the combinatorics of transitive module categories over the monoidal category of finite dimensional <span><math><msub><mrow><mi>sl</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-modules which arise when acting by the latter monoidal category on arbitrary simple <span><math><msub><mrow><mi>sl</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-modules. This gives us a family of eight graphs which can be viewed as <span><math><msub><mrow><mi>sl</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-generalizations of the classical infinite Dynkin diagrams.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 9","pages":"Article 108054"},"PeriodicalIF":0.7,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144687295","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":"Quotient-saturated groups","authors":"Jordi Delgado ,&nbsp;Mallika Roy ,&nbsp;Enric Ventura","doi":"10.1016/j.jpaa.2025.108053","DOIUrl":"10.1016/j.jpaa.2025.108053","url":null,"abstract":"<div><div>We introduce the new notion of <em>quotient-saturation</em> as a measure of the immensity of the quotient structure of a group, and we prove that it is a necessary condition for a non-elementary finitely presented group to embed in a hyperbolic group.</div><div>More generally, we present a sufficient condition — called Congruence Extension Property equipment (in short, CEP-equipment) — for a finitely presented group to be quotient-saturated. Using this property, we deduce that non-elementary finitely presented subgroups of a hyperbolic group (in particular, non-elementary hyperbolic groups themselves) are quotient-saturated.</div><div>Finally, we elaborate on the previous results to extend the scope of CEP-equipment (and hence of quotient-saturation) to finitely presented acylindrically hyperbolic groups.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 9","pages":"Article 108053"},"PeriodicalIF":0.8,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144722515","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":"Cartesian subgroups in graph products of groups","authors":"Fedor Vylegzhanin","doi":"10.1016/j.jpaa.2025.108050","DOIUrl":"10.1016/j.jpaa.2025.108050","url":null,"abstract":"<div><div>The kernel of the natural projection of a graph product of groups onto their direct product is called the Cartesian subgroup of the graph product. This construction generalises commutator subgroups of right-angled Coxeter and Artin groups. Using theory of polyhedral products, we give a lower and an upper bound on the number of relations in presentations of Cartesian groups and on their deficiency. The bounds are related to the fundamental groups of full subcomplexes in the clique complex, and the lower bound coincide with the upper bound if these fundamental groups are free or free abelian.</div><div>Following Li Cai's approach, we also describe an algorithm that computes “small” presentations of Cartesian subgroups.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 9","pages":"Article 108050"},"PeriodicalIF":0.7,"publicationDate":"2025-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144687293","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":"Corrigendum to “Singularities of the Secant Variety” [J. Pure Appl. Algebra 213 (2009) 1129–1132]","authors":"Peter Vermeire","doi":"10.1016/j.jpaa.2025.108048","DOIUrl":"10.1016/j.jpaa.2025.108048","url":null,"abstract":"<div><div>An error in a crucial lemma in the original <span><span>[2]</span></span> directly impacts the main result of the paper. Fortunately, (nearly?) all cases of interest have been corrected in <span><span>[1]</span></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 9","pages":"Article 108048"},"PeriodicalIF":0.7,"publicationDate":"2025-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144633425","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":"The fiber-full scheme","authors":"Yairon Cid-Ruiz ,&nbsp;Ritvik Ramkumar","doi":"10.1016/j.jpaa.2025.108045","DOIUrl":"10.1016/j.jpaa.2025.108045","url":null,"abstract":"<div><div>We introduce the fiber-full scheme which can be seen as the parameter space that generalizes the Hilbert and Quot schemes by controlling the entire cohomological data. Let <span><math><mi>f</mi><mo>:</mo><mi>X</mi><mo>⊂</mo><msubsup><mrow><mi>P</mi></mrow><mrow><mi>S</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo>→</mo><mi>S</mi></math></span> be a projective morphism and <span><math><mi>h</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>h</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>)</mo><mo>:</mo><msup><mrow><mi>Z</mi></mrow><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>→</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> be a fixed tuple of functions. The fiber-full scheme <span><math><msubsup><mrow><mi>Fib</mi></mrow><mrow><mi>F</mi><mo>/</mo><mi>X</mi><mo>/</mo><mi>S</mi></mrow><mrow><mi>h</mi></mrow></msubsup></math></span> is a fine moduli space parametrizing all quotients <span><math><mi>G</mi></math></span> of a fixed coherent sheaf <span><math><mi>F</mi></math></span> on <em>X</em> such that <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msup><msub><mrow><mi>f</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>(</mo><mi>ν</mi><mo>)</mo><mo>)</mo></mrow></math></span> is a locally free <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>S</mi></mrow></msub></math></span>-module of rank equal to <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mi>ν</mi><mo>)</mo></math></span>. In other words, the fiber-full scheme controls the dimension of all cohomologies of all possible twistings, instead of just the Hilbert polynomial. We show that the fiber-full scheme is a quasi-projective <em>S</em>-scheme and a locally closed subscheme of its corresponding Quot scheme. In the context of applications, we demonstrate that the fiber-full scheme provides the natural parameter space for arithmetically Cohen-Macaulay and arithmetically Gorenstein schemes with fixed cohomological data, and for square-free Gröbner degenerations.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 9","pages":"Article 108045"},"PeriodicalIF":0.7,"publicationDate":"2025-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144655013","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":"The Seshadri constants of tangent sheaves on toric varieties","authors":"Chih-Wei Chang","doi":"10.1016/j.jpaa.2025.108046","DOIUrl":"10.1016/j.jpaa.2025.108046","url":null,"abstract":"<div><div>In this paper, we investigate the Seshadri constant <span><math><mi>ε</mi><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>;</mo><mi>p</mi><mo>)</mo></math></span> of the tangent sheaf <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> of a proper <span><math><mi>Q</mi></math></span>-factorial toric variety <em>X</em>. We show that <span><math><mi>ε</mi><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>;</mo><mn>1</mn><mo>)</mo><mo>&gt;</mo><mn>0</mn></math></span> if and only if the following statement holds true: if <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi>v</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mn>0</mn></math></span> where each <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is a positive real number and each <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is the primitive generator of some ray in the fan Δ that defines <em>X</em>, then <span><math><mi>k</mi><mo>≥</mo><mi>dim</mi><mo>⁡</mo><mi>X</mi><mo>+</mo><mn>1</mn></math></span>. Based on the result, we show that a smooth projective toric variety <em>X</em> with <span><math><mi>ε</mi><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>;</mo><mi>p</mi><mo>)</mo><mo>&gt;</mo><mn>0</mn></math></span> for some <span><math><mi>p</mi><mo>∈</mo><mi>X</mi></math></span> is isomorphic to the projective space, confirming a special case of the conjecture proposed by M. Fulger and T. Murayama.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 9","pages":"Article 108046"},"PeriodicalIF":0.7,"publicationDate":"2025-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144655011","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":"Essentials dimension of reductive groups via generically free representations","authors":"Sanghoon Baek,&nbsp;Yeongjong Kim","doi":"10.1016/j.jpaa.2025.108044","DOIUrl":"10.1016/j.jpaa.2025.108044","url":null,"abstract":"<div><div>We provide a simple method to compute upper bounds on the essential dimension of split reductive groups with finite or connected center by means of their generically free representations. Combining our upper bound with previously known lower bound, the exact value of the essential dimension is calculated for some types of reductive groups. As an application, we determine the essential dimension of a semisimple group of classical type or <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>6</mn></mrow></msub></math></span>, and its strict reductive envelope under certain conditions on its center. This extends previous works on simple simply connected groups of type <em>B</em> or <em>D</em> by Brosnan-Reichstein-Vistoli and Chernousov-Merkurjev, strict reductive envelopes of groups of type <em>A</em> by Cernele-Reichstein, and semisimple groups of type <em>B</em> by the authors to any classical type and type <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>6</mn></mrow></msub></math></span> in a uniform way.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 9","pages":"Article 108044"},"PeriodicalIF":0.7,"publicationDate":"2025-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144655012","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}