The Newton polygon of a planar singular curve and its subdivision

TL;DR

This paper explores how the valuations of coefficients of a planar algebraic curve influence the subdivision of its Newton polygon, especially near singular points, and discusses multiple definitions of tropical multiplicity.

Contribution

It introduces a geometric visualization of coefficient constraints on the Newton polygon subdivision related to singular points and compares different notions of tropical multiplicity.

Findings

Identifies faces of the subdivision with total area at least 3/8 m^2 linked to singularity multiplicity

Visualizes coefficient constraints as faces in the Newton polygon subdivision

Discusses three definitions of tropical multiplicity

Abstract

Let a planar algebraic curve $C$ be defined over a valuation field by an equation $F(x,y)=0$. Valuations of the coefficients of $F$ define a subdivision of the Newton polygon $\Delta$ of the curve $C$. If a given point $p$ is of multiplicity $m$ for $C$, then the coefficients of $F$ are subject to certain linear constraints. These constraints can be visualized on the above subdivision of $\Delta$. Namely, we find a distinguished collection of faces of the above subdivision, with total area at least $\frac{3}{8}m^2$. In a sense, the union of these faces in "the region of influence" of the singular point $p$ on the subdivision of $\Delta$. Also, we discuss three different definitions of a tropical point of multiplicity $m$.