The Jacobian conjecture in case of rank or corank less than three
The unproved Jacobian conjecture is the following: if \(F=(F_ 1,\ldots,F_ n):\mathbb{C}^ n\mapsto\mathbb{C}^ n\) is a polynomial mapping with nonzero constant jacobian, then \(F\) is a polynomial automorphism of \(\mathbb{C}^ n\). It was shown by \textit{L. M. Drużkowski} [Math. Ann. 264, 303-313 (1983; Zbl 0504.13006)] that it is sufficient to prove the Jacobian conjecture for every \(n>1\) only for \(F\) having the following form \(F_ j(x_ 1,\ldots,x_ n)=x_ j+(a^ 1_ j+\cdots+a^ n_ jx_ n)^ 3\), \(a^ i_ j\in\mathbb{C}\). The author proves that the Jacobian conjecture holds for \(F\) of the above form in the cases: \(\text{rank}[a^ i_ j]=1, 2\) and \(\text{corank}[a^ i_ j]=1, 2\). Actually, he proves a little more. Namely, in these cases the mapping \(F-F(0)\) is a tame automorphism i.e. a finite composition of linear automorphisms and shears. The latter are polynomial automorphisms of \(\mathbb{C}^ n\) of the form \(T(x_ 1,\ldots,x_ n)=(x_ 1,\ldots,x_{i-1},x_ i+f(x_ 1,\ldots,x_{i- 1},x_{i+1},\ldots, x_ n),x_{i+1},\ldots,x_ n)\).
scientific article; zbMATH DE number 167298
scientific article