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 𝜇 (𝑓) is equal to the Newton number 𝜈 (𝚪₊(𝑓)) of a combinatorial object associated to 𝑓, the Newton polyhedron 𝚪₊ (𝑓). We give a simple condition characterizing, in terms of 𝚪₊ (𝑓) and 𝚪₊ (𝑔), the equality 𝜈 (𝚪₊(𝑓)) = 𝜈 (𝚪₊(𝑔)), for any surface singularities 𝑓 and 𝑔 satisfying 𝚪₊ (𝑓) ⊂ 𝚪₊ (𝑔). This is a complete solution to an Arnold problem (No. 1982-16 in his list of problems) in this case.