A non-negative analogue of the Kouchnirenko formula

Abstract

According to the Kouchnirenko theorem, the Milnor number of an isolated function singularity satisfying certain non-degeneracy condition is equal to an alternating sum (called the Newton number) of the volumes of some polytopes associated with the Newton polyhedron of the singularity. We give a non-negative analogue (without negative summands) of the Kouchnirenko formula and generalize it to a formula for the difference of Milnor numbers of generic singularities with embedded Newton polyhedra. The new formula is supposed to be a hint to the solution of the Arnold's problem on monotonicity of the Newton number. This formula is obtained from the calculation of the asymptotic behavior of critical points of a generic line perturbation of a singularity with a fixed Newton polyhedron, and a new expression for the mixed volume.Comment: 64 pages, 46 figure