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Jacobian Conjecture via Differential Galois Theory
Symmetry, integrability and geometry: methods and applications, Tome 15 (2019) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: polynomial automorphisms, Jacobian problem, strongly normal extensions. 
@article{SIGMA_2019_15_a33,
     author = {El\.zbieta Adamus and Teresa Crespo and Zbigniew Hajto},
     title = {Jacobian {Conjecture} via {Differential} {Galois} {Theory}},
     journal = {Symmetry, integrability and geometry: methods and applications},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a33/}
}
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Elżbieta Adamus; Teresa Crespo; Zbigniew Hajto. Jacobian Conjecture via Differential Galois Theory. Symmetry, integrability and geometry: methods and applications, Tome 15 (2019). http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a33/

[1] Adamus E., Bogdan P., Crespo T., Hajto Z., “An effective study of polynomial maps”, J. Algebra Appl., 16 (2017), 1750141, 13 pp., arXiv: 1506.01654 | DOI | MR | Zbl

[2] Adamus E., Bogdan P., Crespo T., Hajto Z., “Pascal finite polynomial automorphisms”, J. Algebra Appl. (to appear) | DOI

[3] Adamus E., Bogdan P., Hajto Z., “An effective approach to Picard–Vessiot theory and the Jacobian conjecture”, Schedae Informaticae, 26 (2017), 49–59, arXiv: 1506.01662 | DOI

[4] Bass H., Connell E. H., Wright D., “The Jacobian conjecture: reduction of degree and formal expansion of the inverse”, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 287–330 | DOI | MR | Zbl

[5] Campbell L. A., “A condition for a polynomial map to be invertible”, Math. Ann., 205 (1973), 243–248 | DOI | MR

[6] Crespo T., Hajto Z., “Picard–Vessiot theory and the Jacobian problem”, Israel J. Math., 186 (2011), 401–406 | DOI | MR | Zbl

[7] de Bondt M., “Quasi-translations and counterexamples to the homogeneous dependence problem”, Proc. Amer. Math. Soc., 134 (2006), 2849–2856 | DOI | MR | Zbl

[8] de Bondt M., Homogeneous Keller maps, Ph.D. Thesis, Radboud University Nijmegen, 2007 http://webdoc.ubn.ru.nl/mono/b/bondt_m_de/homokema.pdf

[9] Keller O.-H., “Ganze Cremona-Transformationen”, Monatsh. Math. Phys., 47 (1939), 299–306 | DOI | MR

[10] Kovacic J. J., “The differential Galois theory of strongly normal extensions”, Trans. Amer. Math. Soc., 355 (2003), 4475–4522 | DOI | MR | Zbl

[11] Kovacic J. J., “Geometric characterization of strongly normal extensions”, Trans. Amer. Math. Soc., 358 (2006), 4135–4157 | DOI | MR | Zbl

[12] Levelt A. H. M., “Differential Galois theory and tensor products”, Indag. Math. (N.S.), 1 (1990), 439–449 | DOI | MR | Zbl

[13] Smale S., “Mathematical problems for the next century”, Math. Intelligencer, 20 (1998), 7–15 | DOI | MR | Zbl

[14] van den Essen A., “Polynomial automorphisms and the {J}acobian conjecture”, Algèbre Non Commutative, Groupes Quantiques et Invariants (Reims, 1995), v. 2, Sémin. Congr., Soc. Math. France, Paris, 1997, 55–81 | MR

[15] van den Essen A., Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, 190, Birkhäuser Verlag, Basel, 2000 | DOI | MR | Zbl

[16] Wang S. S.-S., “A Jacobian criterion for separability”, J. Algebra, 65 (1980), 453–494 | DOI | MR | Zbl

[17] Yagzhev A. V., “Keller's problem”, Siberian Math. J., 21 (1980), 747–754 | DOI | MR | Zbl