Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 034, 7 pages      arXiv:1901.01566
Contribution to the Special Issue on Algebraic Methods in Dynamical Systems

Jacobian Conjecture via Differential Galois Theory

Elżbieta Adamus a, Teresa Crespo b and Zbigniew Hajto c
a) Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland
b) Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain
c) Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland

Received January 23, 2019, in final form May 01, 2019; Published online May 03, 2019

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.

Key words: polynomial automorphisms; Jacobian problem; strongly normal extensions.

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