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On the set of bad primes in the study of the Casas–Alvero conjecture 论卡萨斯-阿尔维罗猜想研究中的坏素数集
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-04-09 DOI: 10.1007/s40687-024-00444-z
Daniel Schaub, Mark Spivakovsky
{"title":"On the set of bad primes in the study of the Casas–Alvero conjecture","authors":"Daniel Schaub, Mark Spivakovsky","doi":"10.1007/s40687-024-00444-z","DOIUrl":"https://doi.org/10.1007/s40687-024-00444-z","url":null,"abstract":"<p>The Casas–Alvero conjecture predicts that every univariate polynomial over a field of characteristic zero having a common factor with each of its derivatives <span>(H_i(f))</span> is a power of a linear polynomial. One approach to proving the conjecture is to first prove it for polynomials of some small degree <i>d</i>, compile a list of bad primes for that degree (namely, those primes <i>p</i> for which the conjecture fails in degree <i>d</i> and characteristic <i>p</i>) and then deduce the conjecture for all degrees of the form <span>(dp^ell )</span>, <span>(ell in mathbb {N})</span>, where <i>p</i> is a good prime for <i>d</i>. In this paper, we calculate certain distinguished monomials appearing in the resultant <span>(R(f,H_i(f)))</span> and obtain a (non-exhaustive) list of bad primes for every degree <span>(din mathbb {N}setminus {0})</span>.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"20 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140590093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Equivariant cohomology for cyclic groups of square-free order 无平方阶循环群的等变同调
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-03-30 DOI: 10.1007/s40687-024-00443-0
Samik Basu, Surojit Ghosh
{"title":"Equivariant cohomology for cyclic groups of square-free order","authors":"Samik Basu, Surojit Ghosh","doi":"10.1007/s40687-024-00443-0","DOIUrl":"https://doi.org/10.1007/s40687-024-00443-0","url":null,"abstract":"<p>The main objective of this paper is to compute <i>RO</i>(<i>G</i>)-graded cohomology of <i>G</i>-orbits for the group <span>(G=C_n)</span>, where <i>n</i> is a product of distinct primes. We compute these groups for the constant Mackey functor <span>(underline{mathbb {Z}})</span> and the Burnside ring Mackey functor <span>(underline{A})</span>. Among other results, we show that the groups <span>(underline{H}^alpha _G(S^0))</span> are mostly determined by the fixed point dimensions of the virtual representations <span>(alpha )</span>, except in the case of <span>(underline{A})</span> coefficients when the fixed point dimensions of <span>(alpha )</span> have many zeros. In the case of <span>(underline{mathbb {Z}})</span> coefficients, the ring structure on the cohomology is also described. The calculations are then used to prove freeness results for certain <i>G</i>-complexes.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"59 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140590241","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A note on complex plane curve singularities up to diffeomorphism and their rigidity 关于复平面曲线奇异性直至衍射及其刚度的说明
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-03-27 DOI: 10.1007/s40687-024-00439-w
A. Fernández-Hernández, R. Giménez Conejero
{"title":"A note on complex plane curve singularities up to diffeomorphism and their rigidity","authors":"A. Fernández-Hernández, R. Giménez Conejero","doi":"10.1007/s40687-024-00439-w","DOIUrl":"https://doi.org/10.1007/s40687-024-00439-w","url":null,"abstract":"<p>We prove that if two germs of plane curves (<i>C</i>, 0) and <span>((C',0))</span> with at least one singular branch are equivalent by a (real) smooth diffeomorphism, then <i>C</i> is complex isomorphic to <span>(C')</span> or to <span>(overline{C'})</span>. A similar result was shown by Ephraim for irreducible hypersurfaces before, but his proof is not constructive. Indeed, we show that the complex isomorphism is given by the Taylor series of the diffeomorphism. We also prove an analogous result for the case of non-irreducible hypersurfaces containing an irreducible component that is non-factorable. Moreover, we provide a general overview of the different classifications of plane curve singularities.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"56 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140313040","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Euler obstruction, Brasselet number and critical points 欧拉阻塞、布拉塞莱数和临界点
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-03-22 DOI: 10.1007/s40687-024-00426-1
Nicolas Dutertre
{"title":"Euler obstruction, Brasselet number and critical points","authors":"Nicolas Dutertre","doi":"10.1007/s40687-024-00426-1","DOIUrl":"https://doi.org/10.1007/s40687-024-00426-1","url":null,"abstract":"<p>We relate the Brasselet number of a complex analytic function-germ defined on a complex analytic set to the critical points of its real part on the regular locus of the link. Similarly we give a new characterization of the Euler obstruction in terms of the critical points on the regular part of the link of the projection on a generic real line. As a corollary, we obtain a new proof of the relation between the Euler obstruction and the Gauss–Bonnet measure, conjectured by Fu.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"71 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140198123","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Affine homogeneous varieties and suspensions 仿射同质变种和悬浮物
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-03-22 DOI: 10.1007/s40687-024-00438-x
{"title":"Affine homogeneous varieties and suspensions","authors":"","doi":"10.1007/s40687-024-00438-x","DOIUrl":"https://doi.org/10.1007/s40687-024-00438-x","url":null,"abstract":"<h3>Abstract</h3> <p>An algebraic variety <em>X</em> is called a homogeneous variety if the automorphism group <span> <span>({{,textrm{Aut},}}(X))</span> </span> acts on <em>X</em> transitively, and a homogeneous space if there exists a transitive action of an algebraic group on <em>X</em>. We prove a criterion of smoothness of a suspension to construct a wide class of homogeneous varieties. As an application, we give criteria for a Danielewski surface to be a homogeneous variety and a homogeneous space. Also, we construct affine suspensions of arbitrary dimension that are homogeneous varieties but not homogeneous spaces.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"58 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140198116","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Singularities of caustics of surfaces in non-flat Riemannian 4-space form $$^dag $$ 非平坦黎曼 4 空间形式 $$^dag $$ 中曲面的苛求奇点
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-03-16 DOI: 10.1007/s40687-024-00437-y
Liang Chen, Ying Jiang, Haibo Yu
{"title":"Singularities of caustics of surfaces in non-flat Riemannian 4-space form $$^dag $$","authors":"Liang Chen, Ying Jiang, Haibo Yu","doi":"10.1007/s40687-024-00437-y","DOIUrl":"https://doi.org/10.1007/s40687-024-00437-y","url":null,"abstract":"<p>We propose a way to study the caustics of surfaces in non-flat Riemannian 4-space form from the viewpoint of singularity theory in this paper. As an application of the theory of Lagrangian singularity, we study the contact of surfaces with the families of hyperspheres, which is measured by the singularities of functions defined on the surfaces.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"37 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140154988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Reeb spaces of smooth functions on manifolds II 流形上光滑函数的里伯空间 II
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-03-14 DOI: 10.1007/s40687-024-00436-z
Osamu Saeki
{"title":"Reeb spaces of smooth functions on manifolds II","authors":"Osamu Saeki","doi":"10.1007/s40687-024-00436-z","DOIUrl":"https://doi.org/10.1007/s40687-024-00436-z","url":null,"abstract":"<p>The Reeb space of a continuous function on a topological space is the space of connected components of the level sets. In this paper we characterize those smooth functions on closed manifolds whose Reeb spaces have the structure of a finite graph. We also give several explicit examples of smooth functions on closed manifolds such that they themselves or their Reeb spaces have some interesting properties.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"55 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140146360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Synchrony patterns in Laplacian networks 拉普拉斯网络中的同步模式
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-03-14 DOI: 10.1007/s40687-024-00428-z
{"title":"Synchrony patterns in Laplacian networks","authors":"","doi":"10.1007/s40687-024-00428-z","DOIUrl":"https://doi.org/10.1007/s40687-024-00428-z","url":null,"abstract":"<h3>Abstract</h3> <p>A network of coupled dynamical systems is represented by a graph whose vertices represent individual cells and whose edges represent couplings between cells. Motivated by the impact of synchronization results of the Kuramoto networks, we introduce the generalized class of Laplacian networks, governed by mappings whose Jacobian at any point is a symmetric matrix with row entries summing to zero. By recognizing this matrix with a weighted Laplacian of the associated graph, we derive the optimal estimates of its positive, null and negative eigenvalues directly from the graph topology. Furthermore, we provide a characterization of the mappings that define Laplacian networks. Lastly, we discuss stability of equilibria inside synchrony subspaces for two types of Laplacian network on a ring with some extra couplings.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"105 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140146361","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On real algebraic links in the 3-sphere associated with mixed polynomials 论与混合多项式相关的 3 球中的实代数联系
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-03-08 DOI: 10.1007/s40687-024-00424-3
Raimundo N. Araújo dos Santos, Eder L. Sanchez Quiceno
{"title":"On real algebraic links in the 3-sphere associated with mixed polynomials","authors":"Raimundo N. Araújo dos Santos, Eder L. Sanchez Quiceno","doi":"10.1007/s40687-024-00424-3","DOIUrl":"https://doi.org/10.1007/s40687-024-00424-3","url":null,"abstract":"<p>In this paper we construct new classes of mixed singularities that provide realizations of real algebraic links in the 3-sphere. Especially, we describe this construction in the case of semiholomorphic polynomials, which are mixed polynomials that are holomorphic in one variable. Classifications and characterizations of real algebraic links are still open. These new classes of mixed singularities may help to shed light on the Benedetti–Shiota conjecture, which states that any fibered link on the 3-sphere is a real algebraic link.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"54 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140071617","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Bi-Hölder equivalence of real analytic functions 实解析函数的双荷尔德等价性
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-03-07 DOI: 10.1007/s40687-024-00429-y
{"title":"Bi-Hölder equivalence of real analytic functions","authors":"","doi":"10.1007/s40687-024-00429-y","DOIUrl":"https://doi.org/10.1007/s40687-024-00429-y","url":null,"abstract":"<h3>Abstract</h3> <p>In this work, we show that Hölder equivalence of analytic functions germs <span> <span>(({mathbb {R}}^2,0)rightarrow ({mathbb {R}},0))</span> </span> admits continuous moduli.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"35 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140071814","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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