F. J. H. Govantes, W. Mahboub, M. Acosta, M. Spivakovsky
Let $iota:Khookrightarrow Lcong K(x)$ be a simple transcendental extension of valued fields, where $K$ is equipped with a valuation $nu$ of rank 1. That is, we assume given a rank 1 valuation $nu$ of $K$ and its extension $nu'$ to $L$. Let $(R_nu,M_nu,k_nu)$ denote the valuation ring of $nu$. The purpose of this paper is to present a refined version of MacLane's theory of key polynomials, similar to those considered by M. Vaqui'e, and reminiscent of related objects studied by Abhyankar and Moh (approximate roots) and T.C. Kuo. Namely, we associate to $iota$ a countable well ordered set $$ mathbf{Q}={Q_i}_{iinLambda}subset K[x]; $$ the $Q_i$ are called {bf key polynomials}. Key polynomials $Q_i$ which have no immediate predecessor are called {bf limit key polynomials}. Let $beta_i=nu'(Q_i)$. We give an explicit description of the limit key polynomials (which may be viewed as a generalization of the Artin--Schreier polynomials). We also give an upper bound on the order type of the set of key polynomials. Namely, we show that if $operatorname{char} k_nu=0$ then the set of key polynomials has order type at most $omega$, while in the case $operatorname{char} k_nu=p>0$ this order type is bounded above by $omegatimesomega$, where $omega$ stands for the first infinite ordinal.
{"title":"Key polynomials for simple extensions of valued fields","authors":"F. J. H. Govantes, W. Mahboub, M. Acosta, M. Spivakovsky","doi":"10.5427/jsing.2022.25k","DOIUrl":"https://doi.org/10.5427/jsing.2022.25k","url":null,"abstract":"Let $iota:Khookrightarrow Lcong K(x)$ be a simple transcendental extension of valued fields, where $K$ is equipped with a valuation $nu$ of rank 1. That is, we assume given a rank 1 valuation $nu$ of $K$ and its extension $nu'$ to $L$. Let $(R_nu,M_nu,k_nu)$ denote the valuation ring of $nu$. The purpose of this paper is to present a refined version of MacLane's theory of key polynomials, similar to those considered by M. Vaqui'e, and reminiscent of related objects studied by Abhyankar and Moh (approximate roots) and T.C. Kuo. \u0000Namely, we associate to $iota$ a countable well ordered set $$ mathbf{Q}={Q_i}_{iinLambda}subset K[x]; $$ the $Q_i$ are called {bf key polynomials}. Key polynomials $Q_i$ which have no immediate predecessor are called {bf limit key polynomials}. Let $beta_i=nu'(Q_i)$. \u0000We give an explicit description of the limit key polynomials (which may be viewed as a generalization of the Artin--Schreier polynomials). We also give an upper bound on the order type of the set of key polynomials. Namely, we show that if $operatorname{char} k_nu=0$ then the set of key polynomials has order type at most $omega$, while in the case $operatorname{char} k_nu=p>0$ this order type is bounded above by $omegatimesomega$, where $omega$ stands for the first infinite ordinal.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2014-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81738726","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}
We present two results about the relationship between fundamental groups of quasiprojective manifolds and linear systems on a projectivization. We prove the existence of a plane curve with non-abelian fundamental group of the complement which does not admit a mapping onto an orbifold with non-abelian fundamental group. We also find an affine manifold whose irreducible components of its characteristic varieties do not come from the pull-back of the characteristic varieties of an orbifold.
{"title":"On the connection between fundamental groups and pencils with multiple fibers","authors":"Enrique Artal Bartolo, J. I. Cogolludo-Agust'in","doi":"10.5427/jsing.2010.2a","DOIUrl":"https://doi.org/10.5427/jsing.2010.2a","url":null,"abstract":"We present two results about the relationship between fundamental groups of quasiprojective manifolds and linear systems on a projectivization. We prove the existence of a plane curve with non-abelian fundamental group of the complement which does not admit a mapping onto an orbifold with non-abelian fundamental group. We also find an affine manifold whose irreducible components of its characteristic varieties do not come from the pull-back of the characteristic varieties of an orbifold.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2010-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81586940","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}
The distribution of the spectral numbers of an isolated hypersurface singularity is studied in terms of the Bernoulli moments. These are certain rational linear combinations of the higher moments of the spectral numbers. They are related to the generalized Bernoulli polynomials. We conjecture that their signs are alternating and prove this in many cases. One motivation for the Bernoulli moments comes from the comparison with compact complex manifolds.
{"title":"Bernoulli moments of spectral numbers and Hodge numbers<","authors":"Thomas Br'elivet, C. Hertling","doi":"10.5427/jsing.2020.20i","DOIUrl":"https://doi.org/10.5427/jsing.2020.20i","url":null,"abstract":"The distribution of the spectral numbers of an isolated hypersurface singularity is studied in terms of the Bernoulli moments. These are certain rational linear combinations of the higher moments of the spectral numbers. They are related to the generalized Bernoulli polynomials. We conjecture that their signs are alternating and prove this in many cases. One motivation for the Bernoulli moments comes from the comparison with compact complex manifolds.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2004-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90075008","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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