{"title":"Symmetric hyperbolic polynomials","authors":"Grigoriy Blekherman, Julia Lindberg, Kevin Shu","doi":"10.1016/j.jpaa.2025.107869","DOIUrl":"10.1016/j.jpaa.2025.107869","url":null,"abstract":"<div><div>Hyperbolic polynomials have been of recent interest due to applications in a wide variety of fields. We seek to better understand these polynomials in the case when they are symmetric, i.e. invariant under all permutations of the variables. We give a complete characterization of the set of symmetric hyperbolic polynomials of degree 3, and a large class of symmetric hyperbolic polynomials of degree 4. For a class of polynomials, which we call hook-shaped, we relate symmetric hyperbolic polynomials to a class of linear maps of univariate polynomials preserving hyperbolicity, and give evidence toward a beautiful characterization of all such hook-shaped symmetric hyperbolic polynomials. We show that hyperbolicity cones of a class of symmetric hyperbolic polynomials, including all symmetric hyperbolic cubics, are spectrahedral. Finally, we connect testing hyperbolicity of a symmetric polynomial to the degree principle for symmetric nonnegative polynomials.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107869"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162787","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":"Minimal cellular resolutions of powers of matching field ideals","authors":"Oliver Clarke , Fatemeh Mohammadi","doi":"10.1016/j.jpaa.2025.107893","DOIUrl":"10.1016/j.jpaa.2025.107893","url":null,"abstract":"<div><div>We study a family of monomial ideals, called block diagonal matching field ideals, which arise as monomial Gröbner degenerations of determinantal ideals. Our focus is on the minimal free resolutions of these ideals and all of their powers. Initially, we establish their linear quotient property and compute their Betti numbers, illustrating that their minimal free resolution is supported on a regular CW complex. Our proof relies on the results of Herzog and Takayama, demonstrating that ideals with a linear quotient property have a minimal free resolution, and on the construction by Dochtermann and Mohammadi of cellular realizations of these resolutions. We begin by proving the linear quotient property for each power of such an ideal. Subsequently, we show that their corresponding decomposition map is regular, resulting in a minimal cellular resolution. Finally, we demonstrate that distinct decomposition maps lead to different cellular complexes with the same face numbers.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107893"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162789","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 Adams operators on connected graded Hopf algebras","authors":"Y.-Y. Li, G.-S. Zhou","doi":"10.1016/j.jpaa.2025.107877","DOIUrl":"10.1016/j.jpaa.2025.107877","url":null,"abstract":"<div><div>The Adams operators on a Hopf algebra <em>H</em> are the convolution powers of the identity map of <em>H</em>. They are also called Hopf powers or Sweedler powers. It is a natural family of operators on <em>H</em> that contains the antipode. We study the linear properties of the Adams operators when <span><math><mi>H</mi><mo>=</mo><msub><mrow><mo>⨁</mo></mrow><mrow><mi>m</mi><mo>∈</mo><mi>N</mi></mrow></msub><msub><mrow><mi>H</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> is connected graded. The main result is that for any of such <em>H</em>, there exist a PBW type homogeneous basis and a natural total order on it such that the restrictions <span><math><msub><mrow><mi>Ψ</mi></mrow><mrow><mi>n</mi></mrow></msub><msub><mrow><mo>|</mo></mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></msub></math></span> of the Adams operators are simultaneously upper triangularizable with respect to this ordered basis. Moreover, the diagonal coefficients are determined in terms of <em>n</em> and a combinatorial number assigned to the basis elements. As an immediate consequence, we obtain a complete description of the characteristic polynomial of <span><math><msub><mrow><mi>Ψ</mi></mrow><mrow><mi>n</mi></mrow></msub><msub><mrow><mo>|</mo></mrow><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></msub></math></span>, both on eigenvalues and their multiplicities, when <em>H</em> is locally finite and the base field is of characteristic zero. It recovers the main result of the paper <span><span>[2]</span></span> by Aguiar and Lauve, where the approach is different from ours.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107877"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163683","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":"On the embedded versions of degree-2 tropical covers of an elliptic curve","authors":"Numan Amin","doi":"10.1016/j.jpaa.2025.107898","DOIUrl":"10.1016/j.jpaa.2025.107898","url":null,"abstract":"<div><div>We study an equivalent problem to the Hurwitz existence problem in the context of tropical algebraic geometry. For this, we introduced the idea of an algebraic realization of a tropical cover <em>c</em> and homeomorphically faithfulness of the embeddings. In this study, our approach is constructive and we constructed algebraic realizations which are homeomorphically faithful for an arbitrary tropical cover of degree 2 and genus 2 of an abstract elliptic curve. On the basis of length conditions, we divide this into two cases: when the lengths in a tropical cover are equal and when the lengths are unequal. To achieve these results, we also progressed in unfolding and generalized a existing technique to unfold a cycle of certain length under certain conditions.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107898"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143387714","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":"Codimension growth of algebras with superautomorphism","authors":"Antonio Ioppolo , Daniela La Mattina","doi":"10.1016/j.jpaa.2025.107871","DOIUrl":"10.1016/j.jpaa.2025.107871","url":null,"abstract":"<div><div>Let <em>A</em> be a finite dimensional algebra endowed with a superautomorphism over a field of characteristic zero. In this paper we study the asymptotic behavior of the sequence of <em>φ</em>-codimensions <span><math><msubsup><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>φ</mi></mrow></msubsup><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, <span><math><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo></math></span>. More precisely, we shall prove that <span><math><msub><mrow><mi>lim</mi></mrow><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></msub><mo></mo><mroot><mrow><msubsup><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>φ</mi></mrow></msubsup><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mrow><mi>n</mi></mrow></mroot></math></span> always exists and it is an integer related in an explicit way to the dimension of a suitable semisimple subalgebra of <em>A</em>. This result gives a positive answer to a conjecture of Amitsur in this setting. In the final part of the paper we characterize the algebras whose exponential growth is bounded by 2.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107871"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163665","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}
Manohar Kumar , Ramakrishna Nanduri , Kamalesh Saha
{"title":"The slope of the v-function and the Waldschmidt constant","authors":"Manohar Kumar , Ramakrishna Nanduri , Kamalesh Saha","doi":"10.1016/j.jpaa.2025.107881","DOIUrl":"10.1016/j.jpaa.2025.107881","url":null,"abstract":"<div><div>In this paper, we study the asymptotic behavior of the v-number of a Noetherian graded filtration <span><math><mi>I</mi><mo>=</mo><msub><mrow><mo>{</mo><msub><mrow><mi>I</mi></mrow><mrow><mo>[</mo><mi>k</mi><mo>]</mo></mrow></msub><mo>}</mo></mrow><mrow><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span> of a Noetherian <span><math><mi>N</mi></math></span>-graded domain <em>R</em>. Recently, it was shown that <span><math><mi>v</mi><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mo>[</mo><mi>k</mi><mo>]</mo></mrow></msub><mo>)</mo></math></span> is periodically linear in <em>k</em> for <span><math><mi>k</mi><mo>≫</mo><mn>0</mn></math></span>. We show that all these linear functions have the same slope, i.e. <span><math><munder><mi>lim</mi><mrow><mi>k</mi><mo>→</mo><mo>∞</mo></mrow></munder><mo></mo><mfrac><mrow><mi>v</mi><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mo>[</mo><mi>k</mi><mo>]</mo></mrow></msub><mo>)</mo></mrow><mrow><mi>k</mi></mrow></mfrac></math></span> exists, which is equal to <span><math><munder><mi>lim</mi><mrow><mi>k</mi><mo>→</mo><mo>∞</mo></mrow></munder><mo></mo><mfrac><mrow><mi>α</mi><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mo>[</mo><mi>k</mi><mo>]</mo></mrow></msub><mo>)</mo></mrow><mrow><mi>k</mi></mrow></mfrac></math></span>, where <span><math><mi>α</mi><mo>(</mo><mi>I</mi><mo>)</mo></math></span> denotes the minimum degree of a non-zero element in <em>I</em>. In particular, for any Noetherian symbolic filtration <span><math><mi>I</mi><mo>=</mo><msub><mrow><mo>{</mo><msup><mrow><mi>I</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup><mo>}</mo></mrow><mrow><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span> of <em>R</em>, it follows that <span><math><munder><mi>lim</mi><mrow><mi>k</mi><mo>→</mo><mo>∞</mo></mrow></munder><mo></mo><mfrac><mrow><mi>v</mi><mo>(</mo><msup><mrow><mi>I</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup><mo>)</mo></mrow><mrow><mi>k</mi></mrow></mfrac><mo>=</mo><mover><mrow><mi>α</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>(</mo><mi>I</mi><mo>)</mo></math></span>, the Waldschmidt constant of <em>I</em>. Next, for a non-equigenerated square-free monomial ideal <em>I</em>, we prove that <span><math><mi>v</mi><mo>(</mo><msup><mrow><mi>I</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup><mo>)</mo><mo>≤</mo><mi>reg</mi><mo>(</mo><mi>R</mi><mo>/</mo><msup><mrow><mi>I</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup><mo>)</mo></math></span> for <span><math><mi>k</mi><mo>≫</mo><mn>0</mn></math></span>. Also, for an ideal <em>I</em> having the symbolic strong persistence property, we give a linear upper bound on <span><math><mi>v</mi><mo>(</mo><msup><mrow><mi>I</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup><mo>)</mo></math></span>. As an application, we derive some criteria for a square-free monomial ideal <em>I</em> to satisfy <span><math><mi>v</mi><mo>(</mo><msup><mrow><mi>I</mi></mrow><mrow><mo>(</mo><mi>k</mi><","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107881"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162786","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}
David I. Spivak , Richard Garner , Aaron David Fairbanks
{"title":"Functorial aggregation","authors":"David I. Spivak , Richard Garner , Aaron David Fairbanks","doi":"10.1016/j.jpaa.2025.107883","DOIUrl":"10.1016/j.jpaa.2025.107883","url":null,"abstract":"<div><div>We study polynomial comonads and polynomial bicomodules. Polynomial comonads amount to categories. Polynomial bicomodules between categories amount to parametric right adjoint functors between corresponding copresheaf categories. These may themselves be understood as generalized polynomial functors. They are also called data migration functors because of applications in categorical database theory. We investigate several universal constructions in the framed bicategory of categories, retrofunctors, and parametric right adjoints. We then use the theory we develop to model database aggregation alongside querying, all within this rich ecosystem.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107883"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162790","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":"Bloch-Ogus theorem, cyclic homology and deformations of Chow groups","authors":"Sen Yang","doi":"10.1016/j.jpaa.2025.107895","DOIUrl":"10.1016/j.jpaa.2025.107895","url":null,"abstract":"<div><div>Using Bloch-Ogus theorem and Chern character from K-theory to cyclic homology, we answer a question of Green and Griffiths on extending Bloch formula. Moreover, we construct a map from the local Hilbert functor to local cohomology groups. With suitable assumptions, we use this map to answer a question of Bloch on constructing a natural transformation from the local Hilbert functor to cohomological Chow groups.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107895"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162791","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":"Partial groups as symmetric simplicial sets","authors":"Philip Hackney, Justin Lynd","doi":"10.1016/j.jpaa.2025.107864","DOIUrl":"10.1016/j.jpaa.2025.107864","url":null,"abstract":"<div><div>We give a new characterization of partial groups as a subcategory of symmetric (simplicial) sets. This subcategory has an explicit reflection, which permits one to compute colimits in the category of partial groups. We also introduce the notion of a partial groupoid, which encompasses both groupoids and partial groups.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107864"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163606","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":"Flat covers and injective hulls of persistence modules","authors":"Eero Hyry, Ville Puuska","doi":"10.1016/j.jpaa.2025.107874","DOIUrl":"10.1016/j.jpaa.2025.107874","url":null,"abstract":"<div><div>Motivated by recent progress in topological data analysis, we establish a Matlis duality between injective hulls and flat covers of persistence modules. This extends to a duality between minimal flat and minimal injective resolutions. We utilize the theory of flat cotorsion modules and flat covers developed by Enochs and Xu. By means of this theory we can work with persistence modules which are not tame or even pointwise finite-dimensional.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107874"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163679","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}
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