1. Approximate roots, toric resolutions and deformations of a plane branch
- Author
-
Pedro Daniel González Pérez, Departamento de Álgebra [Madrid], Universidad Complutense de Madrid = Complutense University of Madrid [Madrid] (UCM), and Supported by Programa Ramon y Cajal and by MTM2007-6798-C02-02 grants of Ministerio de Educación y Ciencia, Spain.
- Subjects
Power series ,Polynomial ,Quartic plane curve ,Monomial ,Plane curve ,General Mathematics ,2000 MSC Primary 14J17 ,Secondary 32S10, 14M25 ,deformations of a plane curve ,01 natural sciences ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Álgebra ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,14M25 ,Algebraic Geometry (math.AG) ,Principal branch ,32S10 ,Mathematics ,Mathematics::Commutative Algebra ,Plane (geometry) ,010102 general mathematics ,Mathematical analysis ,equisingularity criterion ,approximate roots ,Irreducibility ,010307 mathematical physics ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,14J17 - Abstract
We analyze the expansions in terms of the approximate roots of a Weierstrass polynomial $f$ defining a plane branch $(C,0)$, in the light of the toric embedded resolution of the branch. This leads to the definition of a class of (non equisingular) deformations of a plane branch $(C,0)$ supported on certain monomials in the approximate roots of $f$. As a consequence we find out a Kouchnirenko type formula for the Milnor number $(C,0)$. Our results provide a geometrical approach to Abhyankar's straight line conditions and its consequences. As an application we give an equisingularity criterion for a family of plane curves to be equisingular to a plane branch and we express it algorithmically., Comment: Includes a correction of the previous version of the paper
- Published
- 2010