On Milnor-Orlik's theorem and admissible simultaneous good resolutions

2024/12/09 by Christophe Eyral, Eyral, Christophe, Mutsuo Oka +1
Mathematics · Computer Science · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.2412.06384

Abstract

Let f be a (possibly Newton degenerate) weighted homogeneous polynomial defining an isolated surface singularity at the origin of ℂ³, and let \f_s\ be a generic deformation of its coefficients such that f_s is Newton non-degenerate for s\not=0. We show that there exists an ''admissible'' simultaneous good resolution of the family of functions f_s for all small s, including s=0 which corresponds to the (possibly Newton degenerate) function f. As an application, we give a new geometrical proof of a weak version of the Milnor-Orlik theorem that asserts that the monodromy zeta-function of f (and hence its Milnor number) is completely determined by its weight, its weighted degree and its Newton boundary.

Related

• On the Milnor fibration of certain Newton degenerate functions
• Uniform stable radius and Milnor number for non-degenerate isolated\n complete intersection singularities
• On the monodromy conjecture for non-degenerate hypersurfaces
• Newton polygons and families of polynomials
• Arnold's problem on monotonicity of the Newton number for surface\n singularities