Tropical lifting problem for the intersection of plane curves

TL;DR

This paper investigates when a tropical divisor in the intersection of two tropical plane curves can be realized as the tropicalization of algebraic curve intersections, providing a sufficient condition and an algorithmic approach.

Contribution

It introduces a new sufficient condition involving a graph structure and offers an algorithm to realize tropical divisors as algebraic intersections.

Findings

A specific graph condition ensures realizability of tropical divisors.

An algorithmic method is provided to find algebraic curves matching a given tropical intersection.

The approach advances understanding of the tropical lifting problem for plane curve intersections.

Abstract

Given a tropical divisor $D$ in the intersection of two tropical plane curves, we study when it can be realized as the tropicalization of the intersection of two algebraic curves, and give a sufficient condition. We show that under a certain condition involving a graph determined by these tropical curves, we can algorithmically find algebraic curves such that the tropicalization of their intersection is $D$.