A regularization approach for estimating the type of a plane curve singularity

• Published in: Theoretical computer science Vol. 479; pp. 99 - 119
• Main Authors: Hodorog, Mădălina, Schicho, Josef
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.04.2013
• Subjects: Alexander polynomial; Delta-invariant; Genus; Ill-posed problem; Link of a singularity; Local topological type; Plane curve singularity; Regularization; Symbolic–numeric algorithm; Symbolic–numeric algorithm; Alexander polynomial; Local topological type; Link of a singularity; Delta-invariant; Genus; Ill-posed problem; Regularization; Plane curve singularity
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2012.10.026

We address the algebraic problem of analyzing the local topology of each singularity of a plane complex algebraic curve defined by a squarefree polynomial with both exact (i.e. integers or rationals) and inexact data (i.e. numerical values). For the inexact data, we associate a positive real number that measures the noise in the coefficients. This problem is ill-posed in the sense that tiny changes in the input produce huge changes in the output. We design a regularization method for estimating the local topological type of each singularity of a plane complex algebraic curve. Our regularization method consists of the following: (i) a symbolic–numeric algorithm that computes the approximate local topological type of each singularity; (ii) and a parameter choice rule, i.e. a function in the noise level. We prove that the symbolic–numeric algorithm together with the parameter choice rule computes an approximate solution, which satisfies the convergence for noisy data property. We implement our algorithm in a new software package called GENOM3CK written in the Axel free algebraic geometric modeler and in the Mathemagix free computer algebra system. For our purpose, both of these systems provide modern graphical capabilities, and algebraic and geometric tools for exact and inexact input data.