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Some remarks to the Jacobian conjecture

Lara-Dziembek S., Biernat G., Pawlak E.

Journal of Applied Mathematics and Computational Mechanics

2017

Vol. 16, nr 1

87--96

This work is related to the Jacobian Conjecture. It contains the formulas concerning algebraic dependence of the polynomial mappings having two zeros at infinity and the constant Jacobian. These relations mean that such mappings are non-invertible. They reduce the Jacobian Conjecture only to the case of mappings having one zero at infinity. This case is already solved by Abhyankar. The formulas presented in the paper were illustrated by the large example.

The non-Keller mapping with one zero at infinity

Biernat G., Ciekot A.

Journal of Applied Mathematics and Computational Mechanics

2016

Vol. 15, nr 4

5--10

In this paper the polynomial mapping of two complex variables having one zero at infinity is considered. Unlike with Keller mapping, if determinant of the Jacobian of this mapping is constant then it must be zero.

An example of non-Keller mapping

Pawlak E., Lara-Dziembek S., Biernat G., Woźniakowska M.

Journal of Applied Mathematics and Computational Mechanics

2016

Vol. 15, nr 1

115--121

In the paper a nontrivial example of non-Keller mapping is considered. It is shown that the Jacobian of rare mapping, having one zero at infinity, being constant must vanish.