Arnold's problem on monotonicity of the Newton number for surface singularities

Abstract

According to the Kouchnirenko Theorem, for a generic (meaning non-degenerate in the Kouchnirenko sense) isolated singularity $f$ its Milnor number $\mu (f)$ is equal to the Newton number $\nu (\mathbf{\Gamma}_{+}(f))$ of a combinatorial object associated to $f$, the Newton polyhedron $\mathbf{\Gamma}_+ (f)$. We give a simple condition characterizing, in terms of $\mathbf{\Gamma}_+ (f)$ and $\mathbf{\Gamma}_+ (g)$, the equality $\nu (\mathbf{\Gamma}_{+}(f)) = \nu (\mathbf{\Gamma}_{+}(g))$, for any surface singularities $f$ and $g$ satisfying $\mathbf{\Gamma}_+ (f) \subset \mathbf{\Gamma}_+ (g)$. This is a complete solution to an Arnold problem (No. 1982-16 in his list of problems) in this case.