On a tropical version of the Jacobian conjecture

Dima Grigoriev; Danylo Radchenko

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arXiv · 1902.07733

On a tropical version of the Jacobian conjecture

Dima Grigoriev·Danylo Radchenko

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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.

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Dima Grigoriev, Danylo Radchenko. 2019-02-20. On a tropical version of the Jacobian conjecture. https://arxiv.org/abs/1902.07733

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