Equality of the Singularity Critical Locus Dimension and the Newton Polyhedron Combinatorial Dimension

The purpose of this paper is to determine the local dimension of the critical locus of a generic singularity. We use combinatorial methods to calculate this dimension in terms of a convex object associated with the singularity, called the Newton polyhedron. In this article, we prove that the local dimension of the critical locus of a generic singularity f:(Cn,0)⟶(C,0), n≤4, is equal to the combinatorial dimension of the Newton polyhedron of the gradient mapping ∇f. Therefore, there is some symmetry between combinatorial properties of the Newton polyhedron of a generic singularity and geometric properties of its critical locus.