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The Jacobian conjecture and nilpotent mappings. (Russian) Zbl 1052.14076

Chirka, E. M. (ed.), Complex analysis in modern mathematics. On the 80th anniversary of the birth of Boris Vladimirovich Shabat. Moscow: Izdatel’stvo FAZIS (ISBN 5-7036-0066-9). 167-179 (2001).
Let \(F : \mathbb{C}^n \to \mathbb{C}^n\) be a morphism of complex affine spaces, i.e. in appropiate coordinate systems, \(F\) is given by \(n\) polynomials \(Y_i = F_i (X_1, \ldots, X_n) \in \mathbb{C}^n [X] = \mathbb{C}^n [X_1, \ldots X_n]\) of \(n\) indeterminates. Let \(F'(X) = \det J F (X)\), where \(JF(X)\) is the Jacobi matrix of \(F\).
Jacobian conjecture: JC(n) Let \(F : \mathbb{C}^n \to \mathbb{C}^n\) be a morphism such that \(F'(x) \neq 0\) for all \(x \in \mathbb{C}^n\) (equivalently, \(F'(X) \in \mathbb{C}^*\)). Then \(F\) is an isomorphism, i.e. \(F\) has an inverse which is given also by polynomials.
A morphism \(N = (N_1, \ldots, N_n) : \mathbb{C}^n \to \mathbb{C}^n\) is called {nilpotent}, if its Jacobi matrix \(JN(X)\) is a nilpotent matrix in \(\mathcal{M}(n, \mathbb{C}[X])\).
Nilpotence conjecture: JN(n) A nilpotent morphism \(N : \mathbb{C}^n \to \mathbb{C}^n\) has at most one fixed point.
In this paper, the author proves the following interesting result:
Theorem 2. The Jacobian conjecture JC\((n)\) and the nilpotence conjecture JN\((n)\) are equivalent, i.e. JC\((n)\) is true for all \(n\) iff JN\((n)\) is true for all \(n\). Finally, some particular cases for JN\((n)\) are presented.
For the entire collection see [Zbl 0999.00019].

MSC:

14R15 Jacobian problem
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
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