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Jacobian conjecture via differential Galois theory. (English) Zbl 1420.14132
The authors gives a one-to-one correspondence between the ideals of $$E$$ and the differential ideals of $$K\otimes_CE$$, which is necessary to prove the map $$\Phi:K\otimes_CE\to K\otimes_CR$$ is injective. In addition, they prove that if the map $$\Phi$$ is an isomorphism, then the extension $$K/C$$ is a strongly normal extension. If $$R=C[x]$$ and $K=(C(x),(J^{-1}_F)^T(\frac{\partial}{\partial x_1},\dots,\frac{\partial}{\partial x_n})^T),$ then the Jacobian Conjecture is true in the case that $$\Phi$$ is an isomorphism.
Reviewer: Yan Dan (Changsha)
##### MSC:
 14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) 14R15 Jacobian problem 13N15 Derivations and commutative rings 12F10 Separable extensions, Galois theory
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