## Tokyo Journal of Mathematics

- Tokyo J. Math.
- Volume 42, Number 1 (2019), 51-61.

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

#### 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.

#### Article information

**Source**

Tokyo J. Math., Volume 42, Number 1 (2019), 51-61.

**Dates**

First available in Project Euclid: 18 July 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.tjm/1563436913

**Mathematical Reviews number (MathSciNet)**

MR3982049

**Zentralblatt MATH identifier**

07114900

**Subjects**

Primary: 14T05: Tropical geometry [See also 12K10, 14M25, 14N10, 52B20]

Secondary: 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx] 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx]

#### Citation

TAKAHASHI, Takuhiro. A Note on Tropical Curves and the Newton Diagrams of Plane Curve Singularities. Tokyo J. Math. 42 (2019), no. 1, 51--61. https://projecteuclid.org/euclid.tjm/1563436913