Pub Date : 2025-01-27DOI: 10.1016/j.jalgebra.2025.01.014
Sriram Gopalakrishnan
Let M be an matrix of homogeneous linear forms over a field . If the ideal generated by minors of size is Cohen-Macaulay, then the Gulliksen-Negård complex is a free resolution of . It has recently been shown that by taking into account the syzygy modules for which can be obtained from this complex, one can derive a refined signature-based Gröbner basis algorithm DetGB which avoids reductions to zero when computing a grevlex Gröbner basis for . In this paper, we establish sharp complexity bounds on DetGB. To accomplish this, we prove several results on the sizes of reduced grevlex Gröbner bases of reverse lexicographic ideals, thanks to which we obtain two main complexity results which rely on conjectures similar to that of Fröberg. The first one states that, in the zero-dimensional case, the size of the reduced grevlex Gröbner basis of is bounded from below by asymptotically. The second, also in the zero-dimensional case, states that the complexity of DetGB is bounded from above by asymptotically, where is any complexity exponent for matrix multiplication over .
{"title":"On the arithmetic complexity of computing Gröbner bases of comaximal determinantal ideals","authors":"Sriram Gopalakrishnan","doi":"10.1016/j.jalgebra.2025.01.014","DOIUrl":"10.1016/j.jalgebra.2025.01.014","url":null,"abstract":"<div><div>Let <em>M</em> be an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrix of homogeneous linear forms over a field <span><math><mi>k</mi></math></span>. If the ideal <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></math></span> generated by minors of size <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span> is Cohen-Macaulay, then the Gulliksen-Negård complex is a free resolution of <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></math></span>. It has recently been shown that by taking into account the syzygy modules for <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></math></span> which can be obtained from this complex, one can derive a refined signature-based Gröbner basis algorithm <span>DetGB</span> which avoids reductions to zero when computing a grevlex Gröbner basis for <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></math></span>. In this paper, we establish sharp complexity bounds on <span>DetGB</span>. To accomplish this, we prove several results on the sizes of reduced grevlex Gröbner bases of reverse lexicographic ideals, thanks to which we obtain two main complexity results which rely on conjectures similar to that of Fröberg. The first one states that, in the zero-dimensional case, the size of the reduced grevlex Gröbner basis of <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></math></span> is bounded from below by <span><math><msup><mrow><mi>n</mi></mrow><mrow><mn>6</mn></mrow></msup></math></span> asymptotically. The second, also in the zero-dimensional case, states that the complexity of <span>DetGB</span> is bounded from above by <span><math><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mi>ω</mi><mo>+</mo><mn>3</mn></mrow></msup></math></span> asymptotically, where <span><math><mn>2</mn><mo>≤</mo><mi>ω</mi><mo>≤</mo><mn>3</mn></math></span> is any complexity exponent for matrix multiplication over <span><math><mi>k</mi></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"668 ","pages":"Pages 233-264"},"PeriodicalIF":0.8,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163760","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}
Pub Date : 2025-01-27DOI: 10.1016/j.jalgebra.2025.01.017
Sebastiano Argenti , Mikhail Kochetov , Felipe Yasumura
We classify up to isomorphism the gradings by arbitrary groups on the exceptional classical simple Lie superalgebras , and over an algebraically closed field of characteristic 0. To achieve this, we apply the recent method developed by A. Elduque and M. Kochetov to the known classification of fine gradings up to equivalence on the same superalgebras, which was obtained by C. Draper et al. in 2011. We also classify gradings on the simple Lie superalgebra , whose automorphism group is different from the other members of the A series.
{"title":"Group gradings on exceptional simple Lie superalgebras","authors":"Sebastiano Argenti , Mikhail Kochetov , Felipe Yasumura","doi":"10.1016/j.jalgebra.2025.01.017","DOIUrl":"10.1016/j.jalgebra.2025.01.017","url":null,"abstract":"<div><div>We classify up to isomorphism the gradings by arbitrary groups on the exceptional classical simple Lie superalgebras <span><math><mi>G</mi><mo>(</mo><mn>3</mn><mo>)</mo></math></span>, <span><math><mi>F</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></span> and <span><math><mi>D</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>;</mo><mi>α</mi><mo>)</mo></math></span> over an algebraically closed field of characteristic 0. To achieve this, we apply the recent method developed by A. Elduque and M. Kochetov to the known classification of fine gradings up to equivalence on the same superalgebras, which was obtained by C. Draper et al. in 2011. We also classify gradings on the simple Lie superalgebra <span><math><mi>A</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>, whose automorphism group is different from the other members of the <em>A</em> series.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"668 ","pages":"Pages 447-490"},"PeriodicalIF":0.8,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164540","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}
Pub Date : 2025-01-27DOI: 10.1016/j.jalgebra.2025.01.016
Sathasivam Kalithasan, Tony Nixon Mavely, Viji Zachariah Thomas
We obtain bounds for the size of the Schur multiplier of finite p-groups and finite groups, which improve all existing bounds. Moreover, we obtain bounds for the size of the second cohomology group of a p-group with coefficients in . Denoting the minimal number of generators of a p-group G by , our bound depends on the parameters , , , and . For special p-groups, we further improve our bound when . Moreover, given natural numbers d, δ, k and satisfying and , we construct a capable p-group H of nilpotency class two and exponent p such that the size of the Schur multiplier attains our bound.
{"title":"On the size of the Schur multiplier of finite groups","authors":"Sathasivam Kalithasan, Tony Nixon Mavely, Viji Zachariah Thomas","doi":"10.1016/j.jalgebra.2025.01.016","DOIUrl":"10.1016/j.jalgebra.2025.01.016","url":null,"abstract":"<div><div>We obtain bounds for the size of the Schur multiplier of finite <em>p</em>-groups and finite groups, which improve all existing bounds. Moreover, we obtain bounds for the size of the second cohomology group <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>G</mi><mo>,</mo><mi>Z</mi><mo>/</mo><mi>p</mi><mi>Z</mi><mo>)</mo></math></span> of a <em>p</em>-group with coefficients in <span><math><mi>Z</mi><mo>/</mo><mi>p</mi><mi>Z</mi></math></span>. Denoting the minimal number of generators of a <em>p</em>-group <em>G</em> by <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, our bound depends on the parameters <span><math><mo>|</mo><mi>G</mi><mo>|</mo><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, <span><math><mo>|</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>G</mi><mo>|</mo><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span>, <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>d</mi></math></span>, <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>/</mo><mi>Z</mi><mo>)</mo><mo>=</mo><mi>δ</mi></math></span> and <span><math><mi>d</mi><mo>(</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>G</mi><mo>/</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn>3</mn></mrow></msub><mi>G</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>k</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>. For special <em>p</em>-groups, we further improve our bound when <span><math><mi>δ</mi><mo>−</mo><mn>1</mn><mo>></mo><msup><mrow><mi>k</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>. Moreover, given natural numbers <em>d</em>, <em>δ</em>, <em>k</em> and <span><math><msup><mrow><mi>k</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> satisfying <span><math><mi>k</mi><mo>=</mo><msup><mrow><mi>k</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> and <span><math><mi>δ</mi><mo>−</mo><mn>1</mn><mo>≤</mo><msup><mrow><mi>k</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>, we construct a capable <em>p</em>-group <em>H</em> of nilpotency class two and exponent <em>p</em> such that the size of the Schur multiplier attains our bound.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"668 ","pages":"Pages 420-446"},"PeriodicalIF":0.8,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164158","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}
Pub Date : 2025-01-27DOI: 10.1016/j.jalgebra.2025.01.009
Consuelo Martínez , Efim Zelmanov , Zezhou Zhang
We study the relationship between cyclic homology of Jordan superalgebras and second cohomologies of their Tits-Kantor-Koecher Lie superalgebras.
In particular, we focus on Jordan superalgebras that are Kantor doubles of bracket algebras. The obtained results are applied to computation of second cohomologies and universal central extensions of Hamiltonian and contact type Lie superalgebras over arbitrary rings of coefficients.
{"title":"Cyclic homology of Jordan superalgebras and related Lie superalgebras","authors":"Consuelo Martínez , Efim Zelmanov , Zezhou Zhang","doi":"10.1016/j.jalgebra.2025.01.009","DOIUrl":"10.1016/j.jalgebra.2025.01.009","url":null,"abstract":"<div><div>We study the relationship between cyclic homology of Jordan superalgebras and second cohomologies of their Tits-Kantor-Koecher Lie superalgebras.</div><div>In particular, we focus on Jordan superalgebras that are Kantor doubles of bracket algebras. The obtained results are applied to computation of second cohomologies and universal central extensions of Hamiltonian and contact type Lie superalgebras over arbitrary rings of coefficients.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"668 ","pages":"Pages 600-626"},"PeriodicalIF":0.8,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164537","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}
Pub Date : 2025-01-27DOI: 10.1016/j.jalgebra.2025.01.018
Xiao Han
We construct Hopf bimodules and Yetter-Drinfeld modules of Hopf algebroids as a generalization of the theory for Hopf algebras. More precisely, we show that the categories of Hopf bimodules and Yetter-Drinfeld modules over a Hopf algebroid are equivalent (pre-)braided monoidal categories. Moreover, we also study the duality between finitely generated projective Yetter-Drinfeld modules.
{"title":"Hopf bimodules and Yetter-Drinfeld modules of Hopf algebroids","authors":"Xiao Han","doi":"10.1016/j.jalgebra.2025.01.018","DOIUrl":"10.1016/j.jalgebra.2025.01.018","url":null,"abstract":"<div><div>We construct Hopf bimodules and Yetter-Drinfeld modules of Hopf algebroids as a generalization of the theory for Hopf algebras. More precisely, we show that the categories of Hopf bimodules and Yetter-Drinfeld modules over a Hopf algebroid are equivalent (pre-)braided monoidal categories. Moreover, we also study the duality between finitely generated projective Yetter-Drinfeld modules.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"668 ","pages":"Pages 491-532"},"PeriodicalIF":0.8,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164535","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}
Pub Date : 2025-01-23DOI: 10.1016/j.jalgebra.2024.12.036
Sujeet Dhamore , Amit Hogadi , Rakesh Pawar
We consider the notion of finite type-ness of a site introduced by Morel and Voevodsky, for the étale site of a field. For a given field k, we conjecture that the étale site of is of finite type if and only if the field k admits a finite extension of finite cohomological dimension. We prove this conjecture in some cases, e.g. in the case when k is countable, or in the case when the p-cohomological dimension is infinite for infinitely many primes p.
{"title":"Non-finite type étale sites over fields","authors":"Sujeet Dhamore , Amit Hogadi , Rakesh Pawar","doi":"10.1016/j.jalgebra.2024.12.036","DOIUrl":"10.1016/j.jalgebra.2024.12.036","url":null,"abstract":"<div><div>We consider the notion of finite type-ness of a site introduced by Morel and Voevodsky, for the étale site of a field. For a given field <em>k</em>, we conjecture that the étale site of <span><math><mrow><mi>Sm</mi></mrow><mo>/</mo><mi>k</mi></math></span> is of finite type if and only if the field <em>k</em> admits a finite extension of finite cohomological dimension. We prove this conjecture in some cases, e.g. in the case when <em>k</em> is countable, or in the case when the <em>p</em>-cohomological dimension <span><math><mi>c</mi><msub><mrow><mi>d</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>)</mo></math></span> is infinite for infinitely many primes <em>p</em>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"668 ","pages":"Pages 265-277"},"PeriodicalIF":0.8,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164151","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}
Pub Date : 2025-01-23DOI: 10.1016/j.jalgebra.2025.01.011
Mariko Ohara
Let H be a coFrobenius Hopf algebra over a field k. Let A be a right H-comodule algebra over k.
We recall that the category of right H-comodules admits a certain model structure whose homotopy category is equivalent to the stable category of right H-comodules given in [5]. In the first part of this paper, we show that the category of left A-module objects in admits a model structure, which becomes a model subcategory of the category of -modules endowed with a model structure given in [14] if H is finite dimensional with a certain assumption. Note that is not a Frobenius category in general. We also construct a functorial cofibrant replacement by proceeding the similar argument as in [16].
Hopf-cyclic theory is refered as a theory of cyclic homology of (co)module (co)algebra over a Hopf algebra H whose coefficients in Hopf H-modules. In the latter half of this paper, we see that cyclic H-comodules which give Hopf-cyclic (co)homology with coefficients in Hopf H-modules are contructible in the homotopy category of right H-comodules, and we investigate a Hopf-cyclic (co)homology in slightly modified setting by assuming A a right H-comodule k-Hopf algebra with H-colinear bijective antipode in stable category of right H-comodules and give an analogue of the characteristic map.
{"title":"A model structure and Hopf-cyclic theory on the category of coequivariant modules over a comodule algebra","authors":"Mariko Ohara","doi":"10.1016/j.jalgebra.2025.01.011","DOIUrl":"10.1016/j.jalgebra.2025.01.011","url":null,"abstract":"<div><div>Let <em>H</em> be a coFrobenius Hopf algebra over a field <em>k</em>. Let <em>A</em> be a right <em>H</em>-comodule algebra over <em>k</em>.</div><div>We recall that the category <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>H</mi></mrow></msup></math></span> of right <em>H</em>-comodules admits a certain model structure whose homotopy category is equivalent to the stable category of right <em>H</em>-comodules given in <span><span>[5]</span></span>. In the first part of this paper, we show that the category <span><math><msub><mrow><mi>LMod</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>H</mi></mrow></msup><mo>)</mo></math></span> of left <em>A</em>-module objects in <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>H</mi></mrow></msup></math></span> admits a model structure, which becomes a model subcategory of the category of <span><math><mi>A</mi><mi>#</mi><msup><mrow><mi>H</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-modules endowed with a model structure given in <span><span>[14]</span></span> if <em>H</em> is finite dimensional with a certain assumption. Note that <span><math><msub><mrow><mi>LMod</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>H</mi></mrow></msup><mo>)</mo></math></span> is not a Frobenius category in general. We also construct a functorial cofibrant replacement by proceeding the similar argument as in <span><span>[16]</span></span>.</div><div>Hopf-cyclic theory is refered as a theory of cyclic homology of (co)module (co)algebra over a Hopf algebra <em>H</em> whose coefficients in Hopf <em>H</em>-modules. In the latter half of this paper, we see that cyclic <em>H</em>-comodules which give Hopf-cyclic (co)homology with coefficients in Hopf <em>H</em>-modules are contructible in the homotopy category of right <em>H</em>-comodules, and we investigate a Hopf-cyclic (co)homology in slightly modified setting by assuming <em>A</em> a right <em>H</em>-comodule <em>k</em>-Hopf algebra with <em>H</em>-colinear bijective antipode in stable category of right <em>H</em>-comodules and give an analogue of the characteristic map.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"668 ","pages":"Pages 365-389"},"PeriodicalIF":0.8,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164159","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}
Pub Date : 2025-01-23DOI: 10.1016/j.jalgebra.2024.12.038
Rongmin Zhu , Jiaqun Wei , Zhenxing Di
Let A be a ring. In this paper we study the behavior of silting objects in derived categories of module categories under special base changes with respect to the rings and , where x is a regular element in A. We prove that any silting object in gives rise to silting objects in and after localization and passing to quotient, respectively. On the other hand, it is proved that under some mild conditions, an object in is silting if its corresponding localization and quotient are silting in and , respectively.
{"title":"Silting objects under special base changes","authors":"Rongmin Zhu , Jiaqun Wei , Zhenxing Di","doi":"10.1016/j.jalgebra.2024.12.038","DOIUrl":"10.1016/j.jalgebra.2024.12.038","url":null,"abstract":"<div><div>Let <em>A</em> be a ring. In this paper we study the behavior of silting objects in derived categories of module categories under special base changes with respect to the rings <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>x</mi></mrow></msub></math></span> and <span><math><mi>A</mi><mo>/</mo><mi>x</mi><mi>A</mi></math></span>, where <em>x</em> is a regular element in <em>A</em>. We prove that any silting object in <span><math><mi>D</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> gives rise to silting objects in <span><math><mi>D</mi><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>)</mo></math></span> and <span><math><mi>D</mi><mo>(</mo><mi>A</mi><mo>/</mo><mi>x</mi><mi>A</mi><mo>)</mo></math></span> after localization and passing to quotient, respectively. On the other hand, it is proved that under some mild conditions, an object in <span><math><mi>D</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is silting if its corresponding localization and quotient are silting in <span><math><mi>D</mi><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>)</mo></math></span> and <span><math><mi>D</mi><mo>(</mo><mi>A</mi><mo>/</mo><mi>x</mi><mi>A</mi><mo>)</mo></math></span>, respectively.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"668 ","pages":"Pages 208-232"},"PeriodicalIF":0.8,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164155","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}
Pub Date : 2025-01-23DOI: 10.1016/j.jalgebra.2025.01.010
Sjoerd de Vries
Newton's identities provide a way to express elementary symmetric polynomials in terms of power polynomials over fields of characteristic zero. In this article, we study the failure of this relation in positive characteristic and what can be recovered. In particular, we show how one can write the elementary symmetric polynomials as rational functions in the power polynomials over any commutative unital ring.
{"title":"On Newton's identities in positive characteristic","authors":"Sjoerd de Vries","doi":"10.1016/j.jalgebra.2025.01.010","DOIUrl":"10.1016/j.jalgebra.2025.01.010","url":null,"abstract":"<div><div>Newton's identities provide a way to express elementary symmetric polynomials in terms of power polynomials over fields of characteristic zero. In this article, we study the failure of this relation in positive characteristic and what can be recovered. In particular, we show how one can write the elementary symmetric polynomials as rational functions in the power polynomials over any commutative unital ring.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"668 ","pages":"Pages 348-364"},"PeriodicalIF":0.8,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164160","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}
Pub Date : 2025-01-23DOI: 10.1016/j.jalgebra.2025.01.013
Alexander Olshanskii , Vladimir Shpilrain
The worst-case complexity of group-theoretic algorithms has been studied for a long time. Generic-case complexity, or complexity on random inputs, was introduced and studied relatively recently. In this paper, we address the average-case time complexity of the word problem in several classes of groups and show that it is often the case that the average-case complexity is linear with respect to the length of an input word. The classes of groups that we consider include groups of matrices over rationals (in particular, polycyclic groups), some classes of solvable groups, as well as free products. Along the way, we improve several bounds for the worst-case complexity of the word problem in groups of matrices, in particular in nilpotent groups. For free products, we also address the average-case complexity of the subgroup membership problem and show that it is often linear, too. Finally, we discuss complexity of the identity problem that has not been considered before.
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