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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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