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 the article we prove that the local dimension of the critical locus of a generic singularity \( f:(\mathbb{C}^n,0)\longrightarrow (\mathbb{C},0), n\leq 4, \) is equal to the combinatorial dimension of the Newton polyhedron of the gradient mapping \( \nabla f. \) Therefore there is some symmetry between combinatorial properties of the Newton polyhedron of a generic singularity and geometric properties of its critical locus.