[Paper Review] Newton polygon of a planar singular curve

Nikita Kalinin|arXiv (Cornell University)|Jun 19, 2013

Polynomial and algebraic computation11 references3 citations

TL;DR

This paper studies the Newton polygon subdivision of a planar singular curve over Puiseux series, linking its geometry to the tropicalization of the curve. It establishes quantitative lower bounds on the total area of faces in the subdivision corresponding to vertices or edges of the tropical curve passing through (0,0), showing that if (0,0) lies on an edge, the area is at least $\frac{1}{2}m^2$, and if it is a vertex, the area is at least $\frac{3}{8}m^2$, where $m$ is the multiplicity at (1,1). These results define tropical singularities intrinsically via the subdivision.

ABSTRACT

Consider a planar algebraic curve $C$ over the field $\CC \{\{t\}\}$ of Puiseux series. Suppose $C$ passes through the point $(1,1)$ with multiplicity $m$. The Newton polygon of $C$ is provided with a subdivision dual to the tropicalization $\mathrm{Trop}(C)$ of the curve $C$. I describe properties of such a subdivision. Two of them are the following: a) if the point $(0,0)$ is on an edge of $\mathrm{Trop}(C)$ then the subdivision contains a collection of faces with sum of areas at least $\frac{1}{2}m^2$, these faces correspond to the vertices of the curve $\mathrm{Trop}(C)$ which lie on the maximal long edge passing through the point $(0,0)$; b) if the point $(0,0)$ is a vertex of $\mathrm{Trop}(C)$ then the corresponding sum of areas is at least $\frac{3}{8}m^2$. Also the properties of the subdivision lead to a definition of tropical singularities in pure tropical terms. These results can be formulated for complex algebraic curves with prescribed asymptotics of coefficients of their equations and can be generalized to positive characteristic greater than $m$.

Motivation & Objective

• To understand the structure of the Newton polygon subdivision of a planar algebraic curve defined over the field of Puiseux series.
• To relate geometric features of the curve—specifically its singularities—to the combinatorial structure of its tropicalization.
• To derive quantitative lower bounds on the total area of faces in the Newton polygon subdivision corresponding to tropical curve features passing through (0,0).
• To define tropical singularities in purely combinatorial, tropical terms using the subdivision of the Newton polygon.
• To extend the results to complex algebraic curves with prescribed coefficient asymptotics and to positive characteristic fields where the characteristic exceeds the multiplicity $m$.

Proposed method

• The study uses the Newton polygon of a planar curve $C$ over $\mathbb{C}\{\{t\}\ \/$ to analyze its singular behavior at the point $(1,1)$ with multiplicity $m$.
• The subdivision of the Newton polygon is constructed to be dual to the tropicalization $\mathrm{Trop}(C)$ of the curve $C$, establishing a combinatorial correspondence.
• The analysis focuses on the location of the point $(0,0)$ in the tropical curve: whether it lies on an edge or is a vertex, and how this affects the area of the dual faces in the Newton polygon.
• Key inequalities are derived by relating the multiplicity $m$ at $(1,1)$ to the total area of faces in the Newton polygon subdivision associated with tropical vertices or edges through $(0,0)$.
• The results are generalized to complex algebraic curves with specified asymptotic behavior of coefficients and extended to positive characteristic fields when the characteristic is greater than $m$.
• Tropical singularities are defined via the combinatorial structure of the Newton polygon subdivision, without reference to classical algebraic geometry.

Experimental results

Research questions

• RQ1 What is the minimal total area of faces in the Newton polygon subdivision corresponding to tropical curve vertices lying on the maximal long edge through (0,0) when (0,0) lies on such an edge?
• RQ2 What is the minimal total area of faces in the Newton polygon subdivision corresponding to the vertex at (0,0) in the tropical curve?
• RQ3 How can tropical singularities be defined purely in terms of the combinatorial structure of the Newton polygon subdivision?
• RQ4 How do the asymptotics of the coefficients of a complex algebraic curve's equation affect the structure of its Newton polygon and tropicalization?
• RQ5 In what way can the results over Puiseux series be extended to positive characteristic fields with characteristic greater than $m$?

Key findings

• If the point $(0,0)$ lies on an edge of $\mathrm{Trop}(C)$, then the total area of the corresponding dual faces in the Newton polygon subdivision is at least $\frac{1}{2}m^2$, where $m$ is the multiplicity of the curve at $(1,1)$.
• If the point $(0,0)$ is a vertex of $\mathrm{Trop}(C)$, then the total area of the dual faces in the Newton polygon subdivision is at least $\frac{3}{8}m^2$, again with $m$ being the multiplicity at $(1,1)$.
• The structure of the Newton polygon subdivision provides a complete combinatorial characterization of tropical singularities, enabling their definition in purely tropical terms.
• The results apply to complex algebraic curves with prescribed asymptotics of their coefficient expansions, extending the framework beyond Puiseux series.
• The theory generalizes to positive characteristic fields when the characteristic is greater than $m$, preserving the area bounds and combinatorial structure.