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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 Łojasiewicz exponent at infinity of a polynomial of two real variables

Jankowski P.

Bulletin of the Polish Academy of Sciences. Mathematics

2002

Vol. 50, no 1

25-31

It is shown that the Łojasiewicz exponent at infinity of a positively defined polynomial of two real variables is attained on the set of zeros of a partial derivative of this polynomial. The second result is that the set of the Łojasiewicz exponents at infinity of positively defined polynomials is equal to the set of rational numbers.

Injections into affine hypersurfaces

Jelonek Z.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 1

63-67

Let X be a smooth affine variety of dimension n > 2. Assume that the group H[sub 1](X,Z) is a torsion group and that [chi](X) = 1. Let Y be a projectively smooth affine hypersurface Y [is a subset of] C[sup n+1] of degree d > 1, which is smooth at infinity. Then there is no injective polynomial mapping f : X --> Y. This contradicts a result of Peretz [5].