A Note on Tropical Curves and the Newton Diagrams of Plane Curve Singularities

Source abstract

For an isolated singularity which is Newton non-degenerate and also convenient, the Milnor number can be computed from the complement of its Newton diagram in the first quadrant by using Kouchnirenko's formula. In this paper, we consider tropical curves dual to subdivisions of this complement for a plane curve singularity, and show that there exists a tropical curve by which we can count the Milnor number. Our formula may be regarded as a tropical version of the well-known formula by the real morsification due to A'Campo and Gusein-Zade.