In this paper we discuss an explicit form of divergence-free polynomial derivations in positive characteristic. It involves Jacobian derivations.
本文讨论了正特征多项式无散度导数的一种显式形式。它涉及到雅可比矩阵的推导。
{"title":"A note on divergence-free polynomial derivations in positive characteristic","authors":"Piotr Jędrzejewicz","doi":"10.18778/8142-814-9.10","DOIUrl":"https://doi.org/10.18778/8142-814-9.10","url":null,"abstract":"In this paper we discuss an explicit form of divergence-free polynomial derivations in positive characteristic. It involves Jacobian derivations.","PeriodicalId":273656,"journal":{"name":"Analytic and Algebraic Geometry 3","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125741381","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the article the relation between irreducible curve plane singularities and knots is described. In these terms the topological classification of such singularities is given.
本文描述了不可约曲线平面奇点与结点之间的关系。在这些条件下，给出了奇异点的拓扑分类。
{"title":"Knots of irreducible curve singularities","authors":"T. Krasinski","doi":"10.18778/8142-814-9.11","DOIUrl":"https://doi.org/10.18778/8142-814-9.11","url":null,"abstract":"In the article the relation between irreducible curve plane singularities and knots is described. In these terms the topological classification of such singularities is given.","PeriodicalId":273656,"journal":{"name":"Analytic and Algebraic Geometry 3","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123221187","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note, we explore an apparently new one parameter family of conics associated to a triangle. Given a triangle we study ellipses whose one axis is parallel to one of sides of the triangle. The centers of these ellipses move along three hyperbolas, one for each side of the triangle. These hyperbolas intersect in four common points, which we identify as centers of incircle and the three excircles of the triangle. Thus they belong to a pencil of conics. We trace centers of all conics in the family and establish a surprising fact that they move along the excircle of the triangle. Even though our research is motivated by a problem in elementary geometry, its solution involves some non-trivial algebra and appeal to effective computational methods of algebraic geometry. Our work is illustrated by an animation in Geogebra and accompanied by a Singular file.
{"title":"A family of hyperbolas associated to a triangle","authors":"Maciej Zięba","doi":"10.18778/8142-814-9.17","DOIUrl":"https://doi.org/10.18778/8142-814-9.17","url":null,"abstract":"In this note, we explore an apparently new one parameter family of conics associated to a triangle. Given a triangle we study ellipses whose one axis is parallel to one of sides of the triangle. The centers of these ellipses move along three hyperbolas, one for each side of the triangle. These hyperbolas intersect in four common points, which we identify as centers of incircle and the three excircles of the triangle. Thus they belong to a pencil of conics. We trace centers of all conics in the family and establish a surprising fact that they move along the excircle of the triangle. Even though our research is motivated by a problem in elementary geometry, its solution involves some non-trivial algebra and appeal to effective computational methods of algebraic geometry. Our work is illustrated by an animation in Geogebra and accompanied by a Singular file.","PeriodicalId":273656,"journal":{"name":"Analytic and Algebraic Geometry 3","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123791257","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Points and lines can be regarded as the simplest geometrical objects. Incidence relations between them have been studied since ancient times. Strangely enough our knowledge of this area of mathematics is still far from being complete. In fact a number of interesting and apparently difficult conjectures has been raised just recently. Additionally a number of interesting connections to other branches of mathematics have been established. This is an attempt to record some of these recent developments.
{"title":"A few introductory remarks on line arrangements","authors":"J. Szpond","doi":"10.18778/8142-814-9.15","DOIUrl":"https://doi.org/10.18778/8142-814-9.15","url":null,"abstract":"Points and lines can be regarded as the simplest geometrical objects. Incidence relations between them have been studied since ancient times. Strangely enough our knowledge of this area of mathematics is still far from being complete. In fact a number of interesting and apparently difficult conjectures has been raised just recently. Additionally a number of interesting connections to other branches of mathematics have been established. This is an attempt to record some of these recent developments.","PeriodicalId":273656,"journal":{"name":"Analytic and Algebraic Geometry 3","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123410575","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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