Computation of Milnor numbers and critical values at infinity

2003/09/19 by Arnaud Bodin
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #math.AG #math.GT

paper · pdf · doi:10.1016/j.jsc.2004.04.003

published as Journal of Symbolic Computation (2004, vol. 38, 1417-1427) · 9 pages.To download the libraries for Singular see http://www-gat.univ-lille1.fr/~bodin/

arxiv created 2003/09/19 · openalex publication_date 2004/07/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We describe how to compute topological objects associated to a polynomial map of several complex variables with isolated singularities. These objects are: the affine critical values, the affine Milnor numbers for all irregular fibers, the critical values at infinity, and the Milnor numbers at infinity for all irregular fibers. Then for a family of polynomials we detect parameters where the topology of the polynomials can change. Implementation and examples are given with the computer algebra system Singular.

Related

• Invariance of Milnor numbers and topology of complex polynomials
• A generalization of Milnor's formula
• Hypersurface Singularities and Milnor Equisingularity
• Unfolding polynomial maps at infinity
• A Formula For the Homology Groups of the Milnor Fibres of an Isolated Real Hypersurface Singularity