Let 𝕂 denote ℝ or ℂ, n > 1. The Jacobian Conjecture can be formulated as follows: If F:𝕂ⁿ → 𝕂ⁿ is a polynomial map with a constant nonzero jacobian, then F is a polynomial automorphism. Although the Jacobian Conjecture is still unsolved even in the case n = 2, it is convenient to consider the so-called Generalized Jacobian Conjecture (for short (GJC)): the Jacobian Conjecture holds for every n>1. We present the reduction of (GJC) to the case of F of degree 3 and of symmetric homogeneous form and prove (JC) for maps having cubic linear form with symmetric F'(x), more precisely: polynomial maps of cubic linear form with symmetric F'(x) and constant nonzero jacobian are tame automorphisms.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap87-0-7,
author = {Ludwik M. Dru\.zkowski},
title = {The Jacobian Conjecture: symmetric reduction and solution in the symmetric cubic linear case},
journal = {Annales Polonici Mathematici},
volume = {85},
year = {2005},
pages = {83-92},
zbl = {1098.14047},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap87-0-7}
}
Ludwik M. Drużkowski. The Jacobian Conjecture: symmetric reduction and solution in the symmetric cubic linear case. Annales Polonici Mathematici, Tome 85 (2005) pp. 83-92. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap87-0-7/