The Jacobian Conjecture: symmetric reduction and solution in the symmetric cubic linear case
Volume 87 / 2005
Abstract
Let denote \mathbb R or \mathbb C, n>1. The Jacobian Conjecture can be formulated as follows: If F:{\mathbb K}^n \to {\mathbb K}^n 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.