# Jacobian Conjecture via Differential Galois Theory

## TL;DR

This paper establishes a novel criterion for polynomial map invertibility using differential Galois theory, connecting algebraic invertibility with properties of differential field extensions.

## Contribution

It introduces a new approach to the Jacobian Conjecture by linking polynomial invertibility to differential Galois theory and Picard-Vessiot extensions.

## Key findings

- Polynomial maps are invertible iff associated differential ring homomorphisms are bijective.
- Uses Picard-Vessiot extensions to characterize invertibility.
- Provides a differential Galois theoretic perspective on the Jacobian Conjecture.

## Abstract

We prove that a polynomial map is invertible if and only if some associated differential ring homomorphism is bijective. To this end, we use a theorem of Crespo and Hajto linking the invertibility of polynomial maps with Picard-Vessiot extensions of partial differential fields, the theory of strongly normal extensions as presented by Kovacic and the characterization of Picard-Vessiot extensions in terms of tensor products given by Levelt.