The number of vertices of a tropical curve is bounded by its area

TL;DR

This paper establishes a fundamental bound on the number of vertices of a tropical curve based on its area, linking geometric size to combinatorial complexity.

Contribution

It introduces the concept of tropical area and proves that the number of vertices is bounded by this area, using elementary methods inspired by algebraic intersection theory.

Findings

Number of vertices is bounded by tropical area

Tropical area is a new geometric invariant

Moduli space of bounded-area tropical curves is finite

Abstract

We introduce the notion of tropical area of a tropical curve defined in an open subset of $\mathbb R^n$. We prove that the number of vertices of a tropical curve is bounded by the area of the curve. The approach is totally elementary yet tricky. Our proof employs ideas from intersection theory in algebraic geometry. The result can be interpreted as the fact that the moduli space of tropical curves with bounded area is of finite type.