# On a tropical version of the Jacobian conjecture

## TL;DR

This paper explores conditions under which tropical rational and polynomial maps are isomorphisms, establishing criteria based on Jacobian matrices and their properties, thus providing tropical analogs to classical Jacobian conjecture results.

## Contribution

It introduces tropical conditions involving Jacobian matrices that guarantee a tropical rational or polynomial map is an isomorphism, extending classical conjectures into tropical geometry.

## Key findings

- Convex hull of Jacobians not containing singular matrices implies isomorphism.
- Tropical polynomial maps with all Jacobians of same sign are isomorphisms.
- Preimage singleton condition with same sign Jacobians ensures isomorphism.

## Abstract

We prove for a tropical rational map that if for any point the convex hull of Jacobian matrices at smooth points in a neighborhood of the point does not contain singular matrices then the map is an isomorphism. We also show that a tropical polynomial map on the plane is an isomorphism if all the Jacobians have the same sign (positive or negative). In addition, for a tropical rational map we prove that if the Jacobians have the same sign and if its preimage is a singleton at least at one regular point then the map is an isomorphism.