2022/07/04 by R. Giménez Conejero, Conejero, R. Giménez, J. J. Nuño‐Ballesteros +1
Computer Science · Mathematics · Engineering · #semigroups and automata theory #Advanced Topology and Set Theory #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2207.01735
We prove that a map germ f:(ℂn,S)→(ℂn+1,0) with isolated instability is stable if and only if μI(f)=0, where μI(f) is the image Milnor number defined by Mond. In a previous paper we proved this result with the additional assumption that f has corank one. The proof here is also valid for corank ≥ 2, provided that (n,n+1) are nice dimensions in Mather's sense (so μI(f) is well defined). Our result can be seen as a weak version of a conjecture by Mond, which says that the Ae-codimension of f is ≤ μI(f), with equality if f is weighted homogeneous. As an application, we deduce that the bifurcation set of a versal unfolding of f is a hypersurface.