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

We address the algebraic problem of analysing 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