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

摘要

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