The Newton polygon of a rational plane curve

TL;DR

This paper explicitly determines the Newton polygon of a rational plane curve's implicit equation using parametrization multiplicities, providing new geometric insights and applications to generic parametrizations.

Contribution

It offers an intersection-theoretical proof linking Newton polygons to parametrization multiplicities and characterizes the space of rational curves with a given Newton polygon.

Findings

Explicit formula for Newton polygon from parametrization multiplicities

Unirationality and dimension of the variety of rational curves with fixed Newton polygon

Any convex lattice polygon with positive area is realizable as a Newton polygon of a rational curve

Abstract

The Newton polygon of the implicit equation of a rational plane curve is explicitly determined by the multiplicities of any of its parametrizations. We give an intersection-theoretical proof of this fact based on a refinement of the Kushnirenko-Bernstein theorem. We apply this result to the determination of the Newton polygon of a curve parameterized by generic Laurent polynomials or by generic rational functions, with explicit genericity conditions. We also show that the variety of rational curves with given Newton polygon is unirational and we compute its dimension. As a consequence, we obtain that any convex lattice polygon with positive area is the Newton polygon of a rational plane curve.