Plucker Formulas for Plane Algebraic Curves with a Given Newton Polygon

TL;DR

This paper computes the number of inflection points and bitangents of generic plane algebraic curves with a specified Newton polygon, and shows the dual curve's singularities are limited to nodes and cusps for large polygons.

Contribution

It introduces explicit formulas for inflection points and bitangents based on the Newton polygon and proves the dual curve's singularity structure for large polygons.

Findings

Number of inflection points and bitangents computed

Dual curve has only nodes and cusps for large polygons

Provides a link between Newton polygons and curve singularities

Abstract

Let $C$ be a generic complex plane plane curve with a given Newton polygon $P$. We compute the number of its inflection points and bitangents (equivalently, the number of singularities of the projectively dual curve $C^\vee$). We also prove that $C^\vee$ has no singularities other than nodes and cusps for large enough polygons $P$.