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According to the Kouchnirenko theorem, for a generic (preciselynon-degenerate in the Kouchnirenko sense) isolated singularity $f$ its Milnornumber $\mu (f)$ is equal to the Newton number $\nu (\Gamma_{+}(f))$ of acombinatorial 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 $\nu (\Gamma_{+}(f)) = \nu (\Gamma_{+}(g))$, forany surface singularities $f$ and $g$ satisfying $\Gamma_+ (f) \subset \Gamma_+(g)$. This is a complete solution to an Arnold's problem (1982-16) in thiscase.

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