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On a tropical version of the Jacobian conjecture

Research paper by Dima Grigoriev, Danylo Radchenko

Indexed on: 20 Feb '19Published on: 20 Feb '19Published in: arXiv - Mathematics - Algebraic Geometry



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.