The Jacobian Conjecture: symmetric reduction and solution in the symmetric cubic linear case
Volume 87 / 2005
Abstract
Let $\mathbb K$ 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.
