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

Abstract

According to the Kouchnirenko theorem, for a generic (precisely non-degenerate in the Kouchnirenko sense) isolated singularity $f$ its Milnor number $\mu (f)$ is equal to the Newton number $\nu (\Gamma_{+}(f))$ of a combinatorial object associated to $f$, the Newton polyhedron $\Gamma_+ (f)$. We give a simple condition characterising, in terms of $\Gamma_+ (f)$ and $\Gamma_+ (g)$, the equality $\nu (\Gamma_{+}(f)) = \nu (\Gamma_{+}(g))$, for any surface singularities $f$ and $g$ satisfying $\Gamma_+ (f) \subset \Gamma_+ (g)$. This is a complete solution to an Arnold's problem (1982-16) in this case.