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.

12 pages, 9 figures