A note on tropical curves and the Newton diagrams of plane curve singularities

TL;DR

This paper explores the relationship between tropical curves and Newton diagrams of plane curve singularities, establishing a new formula connecting tropical geometry with classical singularity invariants.

Contribution

It introduces a novel tropical curve construction dual to Newton diagram subdivisions, linking tropical geometry with singularity invariants like the Milnor number.

Findings

Existence of a tropical curve satisfying a specific formula

Connection between tropical curves and classical singularity formulas

Extension of known formulas to tropical geometric context

Abstract

For a convenient and Newton non-degenerate singularity, the Milnor number is computed from the complement of its Newton diagram in the first quadrant, so-called Kouchnirenko's formula. In this paper, we consider tropical curves dual to subdivisions of this complement for a plane curve singularity and show the existence of a tropical curve satisfying a certain formula, which looks like a well-known formula for a real morsification due to A'Campo and Gusein-Zade.