×

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
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adamus, El\.{z}bieta and Bogdan, Pawe{\l} and Hajto, Zbigniew, An effective approach to {P}icard–{V}essiot theory and the {J}acobian conjecture, Schedae Informaticae, 26, 1, 49-59, (2017) · Zbl 1163.90353 · doi:10.4467/20838476SI.17.004.8150
[2] Adamus, El\.{z}bieta and Bogdan, Pawe{\l} and Crespo, Teresa and Hajto, Zbigniew, An effective study of polynomial maps, Journal of Algebra and its Applications, 16, 8, 1750141, 13 pages, (2017) · Zbl 1163.90353 · doi:10.1142/S0219498817501419
[3] Adamus, El\.{z}bieta and Bogdan, Pawe{\l} and Crespo, Teresa and Hajto, Zbigniew, Pascal finite polynomial automorphisms, Journal of Algebra and its Applications · Zbl 1163.90353 · doi:10.1142/S021949881950124X
[4] Bass, Hyman and Connell, Edwin H. and Wright, David, The {J}acobian conjecture: reduction of degree and formal expansion of the inverse, American Mathematical Society. Bulletin. New Series, 7, 2, 287-330, (1982) · Zbl 1163.90353 · doi:10.1090/S0273-0979-1982-15032-7
[5] de Bondt, Michiel, Homogeneous {K}eller maps · Zbl 1163.90353
[6] de Bondt, Michiel, Quasi-translations and counterexamples to the homogeneous dependence problem, Proceedings of the American Mathematical Society, 134, 10, 2849-2856, (2006) · Zbl 1163.90353 · doi:10.1090/S0002-9939-06-08335-3
[7] Campbell, L. Andrew, A condition for a polynomial map to be invertible, Mathematische Annalen, 205, 243-248, (1973) · Zbl 1163.90353 · doi:10.1007/BF01349234
[8] Crespo, Teresa and Hajto, Zbigniew, Picard–{V}essiot theory and the {J}acobian problem, Israel Journal of Mathematics, 186, 401-406, (2011) · Zbl 1163.90353 · doi:10.1007/s11856-011-0145-y
[9] van den Essen, Arno, Polynomial automorphisms and the {J}acobian conjecture, Progress in Mathematics, 190, xviii+329, (2000), Birkh\"{a}user Verlag, Basel · Zbl 1163.90353 · doi:10.1007/978-3-0348-8440-2
[10] van den Essen, Arno, Polynomial automorphisms and the {J}acobian conjecture, Alg\`ebre Non Commutative, Groupes Quantiques et Invariants ({R}eims, 1995), S\'{e}min. Congr., 2, 55-81, (1997), Soc. Math. France, Paris · Zbl 1163.90353
[11] Keller, Ott-Heinrich, Ganze {C}remona-{T}ransformationen, Monatshefte f\"{u}r Mathematik und Physik, 47, 1, 299-306, (1939) · Zbl 1163.90353 · doi:10.1007/BF01695502
[12] Kovacic, Jerald J., The differential {G}alois theory of strongly normal extensions, Transactions of the American Mathematical Society, 355, 11, 4475-4522, (2003) · Zbl 1163.90353 · doi:10.1090/S0002-9947-03-03306-3
[13] Kovacic, Jerald J., Geometric characterization of strongly normal extensions, Transactions of the American Mathematical Society, 358, 9, 4135-4157, (2006) · Zbl 1163.90353 · doi:10.1090/S0002-9947-06-03868-2
[14] Levelt, A. H. M., Differential {G}alois theory and tensor products, Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Mathematicae. New Series, 1, 4, 439-449, (1990) · Zbl 1163.90353 · doi:10.1016/0019-3577(90)90012-C
[15] Smale, Steve, Mathematical problems for the next century, The Mathematical Intelligencer, 20, 2, 7-15, (1998) · Zbl 1163.90353 · doi:10.1007/BF03025291
[16] Wang, Stuart Sui-Sheng, A {J}acobian criterion for separability, Journal of Algebra, 65, 2, 453-494, (1980) · Zbl 1163.90353 · doi:10.1016/0021-8693(80)90233-1
[17] Yagzhev, A. V., Keller’s problem, Siberian Mathematical Journal, 21, 5, 747-754, (1980) · Zbl 1163.90353 · doi:10.1007/BF00973892
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.