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Saddle point braids of braided fibrations and pseudo-fibrations 辫状纤维和伪纤维的鞍点辫状结构
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-04-17 DOI: 10.1007/s40687-024-00446-x
Benjamin Bode, Mikami Hirasawa
{"title":"Saddle point braids of braided fibrations and pseudo-fibrations","authors":"Benjamin Bode, Mikami Hirasawa","doi":"10.1007/s40687-024-00446-x","DOIUrl":"https://doi.org/10.1007/s40687-024-00446-x","url":null,"abstract":"<p>Let <span>(g_t)</span> be a loop in the space of monic complex polynomials in one variable of fixed degree <i>n</i>. If the roots of <span>(g_t)</span> are distinct for all <i>t</i>, they form a braid <span>(B_1)</span> on <i>n</i> strands. Likewise, if the critical points of <span>(g_t)</span> are distinct for all <i>t</i>, they form a braid <span>(B_2)</span> on <span>(n-1)</span> strands. In this paper we study the relationship between <span>(B_1)</span> and <span>(B_2)</span>. Composing the polynomials <span>(g_t)</span> with the argument map defines a pseudo-fibration map on the complement of the closure of <span>(B_1)</span> in <span>({mathbb {C}}times S^1)</span>, whose critical points lie on <span>(B_2)</span>. We prove that for <span>(B_1)</span> a T-homogeneous braid and <span>(B_2)</span> the trivial braid this map can be taken to be a fibration map. In the case of homogeneous braids we present a visualization of this fact. Our work implies that for every pair of links <span>(L_1)</span> and <span>(L_2)</span> there is a mixed polynomial <span>(f:{mathbb {C}}^2rightarrow {mathbb {C}})</span> in complex variables <i>u</i>, <i>v</i> and the complex conjugate <span>(overline{v})</span> such that both <i>f</i> and the derivative <span>(f_u)</span> have a weakly isolated singularity at the origin with <span>(L_1)</span> as the link of the singularity of <i>f</i> and <span>(L_2)</span> as a sublink of the link of the singularity of <span>(f_u)</span>.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"9 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140610487","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
The crossing numbers of amphicheiral knots 双螺旋结的交叉数
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-04-17 DOI: 10.1007/s40687-024-00440-3
A. Stoimenow
{"title":"The crossing numbers of amphicheiral knots","authors":"A. Stoimenow","doi":"10.1007/s40687-024-00440-3","DOIUrl":"https://doi.org/10.1007/s40687-024-00440-3","url":null,"abstract":"<p>We determine the crossing numbers of (prime) amphicheiral knots. This problem dates back to the origin of knot tables by Tait and Little at the end of the nineteenth century. The proof is the most substantial application of the semiadequacy formulas for the edge coefficients of the Jones polynomial.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"53 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140610492","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
The generic equivalence among the Lipschitz saturations of a sheaf of modules 模块 Sheaf 的 Lipschitz 饱和度之间的一般等价关系
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-04-12 DOI: 10.1007/s40687-024-00442-1
Terence James Gaffney, Thiago Filipe da Silva
{"title":"The generic equivalence among the Lipschitz saturations of a sheaf of modules","authors":"Terence James Gaffney, Thiago Filipe da Silva","doi":"10.1007/s40687-024-00442-1","DOIUrl":"https://doi.org/10.1007/s40687-024-00442-1","url":null,"abstract":"<p>In this work, we extend the concept of the Lipschitz saturation of an ideal to the context of modules in some different ways, and we prove they are generically equivalent.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"40 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140590092","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
Dual relations between line congruences in $$mathbb {R}^3$$ and surfaces in $$mathbb {R}^4$$ $$mathbb {R}^3$$ 中的线全等与 $$mathbb {R}^4$$ 中的面之间的双重关系
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-04-09 DOI: 10.1007/s40687-024-00445-y
Marcos Craizer, Ronaldo Garcia
{"title":"Dual relations between line congruences in $$mathbb {R}^3$$ and surfaces in $$mathbb {R}^4$$","authors":"Marcos Craizer, Ronaldo Garcia","doi":"10.1007/s40687-024-00445-y","DOIUrl":"https://doi.org/10.1007/s40687-024-00445-y","url":null,"abstract":"<p>There is a natural duality between line congruences in <span>(mathbb {R}^3)</span> and surfaces in <span>(mathbb {R}^4)</span> that sends principal lines into asymptotic lines. The same correspondence takes the discriminant curve of a line congruence into the parabolic curve of the dual surface. Moreover, it takes the ridge curves to the flat ridge curves, while the subparabolic curves of a line congruence are taken to certain curves on the surface that we call flat subparabolic curves. In this paper, we discuss these relations and describe the generic behavior of the subparabolic curves at the discriminant curve of the line congruence, or equivalently, the parabolic curve of the dual surface. We also discuss Loewner’s conjectures under the duality viewpoint.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"9 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140590087","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 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
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