The Jacobian Conjecture: symmetric reduction and solution in the symmetric cubic linear case

Ludwik M. Drużkowski. "The Jacobian Conjecture: symmetric reduction and solution in the symmetric cubic linear case." Annales Polonici Mathematici 87.1 (2005): 83-92. <http://eudml.org/doc/280721>.

@article{LudwikM2005,
abstract = {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.},
author = {Ludwik M. Drużkowski},
journal = {Annales Polonici Mathematici},
keywords = {symmetric homogeneous form; tame automorphism},
language = {eng},
number = {1},
pages = {83-92},
title = {The Jacobian Conjecture: symmetric reduction and solution in the symmetric cubic linear case},
url = {http://eudml.org/doc/280721},
volume = {87},
year = {2005},
}

TY - JOUR
AU - Ludwik M. Drużkowski
TI - The Jacobian Conjecture: symmetric reduction and solution in the symmetric cubic linear case
JO - Annales Polonici Mathematici
PY - 2005
VL - 87
IS - 1
SP - 83
EP - 92
AB - 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.
LA - eng
KW - symmetric homogeneous form; tame automorphism
UR - http://eudml.org/doc/280721
ER -