Jacobian conjecture via differential Galois theory

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