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The Jacobian conjecture and nilpotent maps. (English. Russian original) Zbl 1011.14020

J. Math. Sci., New York 106, No. 5, 3312-3319 (2001); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 70, VINITI, Moscow 120-133 (2001).
The paper is devoted to the next equivalent condition to the Jacobian conjecture. A polynomial mapping \(N=(N_1,\dots ,N_n):\mathbb C^n \to \mathbb C^n\) is called nilpotent iff its jacobian matrix is nilpotent, i.e. \(\Biggl(\frac{\partial N_i}{\partial X_j}\Biggr)^m=0\).
The author proves that the Jacobian conjecture for every \(n\) is equivalent to the fact that for every \(n\) any nilpotent mapping of \(\mathbb C^n\) has at most one fixed point. He also verifies this new conjecture in some particular classes of mappings.

MSC:

14R15 Jacobian problem
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13B10 Morphisms of commutative rings
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