Jacobian Conjecture via Differential Galois Theory

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.