Polynomial Algorithms in Computer Algebra
Polynomial Algorithms in Computer Algebra
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
Subdivision methods for solving polynomial equations
Journal of Symbolic Computation
Approximate parametrization of plane algebraic curves by linear systems of curves
Computer Aided Geometric Design
GENOM3CK: a library for genus computation of plane complex algebraic curves using knot theory
ACM Communications in Computer Algebra
A Symbolic-Numeric Algorithm for Computing the Alexander Polynomial of a Plane Curve Singularity
SYNASC '10 Proceedings of the 2010 12th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
A regularization approach for estimating the type of a plane curve singularity
Theoretical Computer Science
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We approach the algebraic problem of computing topological invariants for the singularities of a plane complex algebraic curve defined by a squarefree polynomial with inexactly-known coefficients. Consequently, we deal with an ill-posed problem in the sense that, tiny changes in the input data lead to dramatic modifications in the output solution. We present a regularization method for handling the illposedness of the problem. For this purpose, we first design symbolic-numeric algorithms to extract structural information on the plane complex algebraic curve: (i) we compute the link of each singularity by numerical equation solving; (ii) we compute the Alexander polynomial of each link by using algorithms from computational geometry and combinatorial objects from knot theory; (iii) we derive a formula for the delta-invariant and the genus. We then prove the convergence for inexact data of the symbolic-numeric algorithms by using concepts from algebraic geometry and topology. Moreover we perform several numerical experiments, which support the validity for the convergence statement.