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A counterexample to a conjecture of Drużkowski and Rusek

Arno van den Essen (1995)

Annales Polonici Mathematici

Let F = X + H be a cubic homogeneous polynomial automorphism from n to n . Let p be the nilpotence index of the Jacobian matrix JH. It was conjectured by Drużkowski and Rusek in [4] that d e g F - 1 3 p - 1 . We show that the conjecture is true if n ≤ 4 and false if n ≥ 5.

A geometric approach to the Jacobian Conjecture in ℂ²

Ludwik M. Drużkowski (1991)

Annales Polonici Mathematici

We consider polynomial mappings (f,g) of ℂ² with constant nontrivial jacobian. Using the Riemann-Hurwitz relation we prove among other things the following: If g - c (resp. f - c) has at most two branches at infinity for infinitely many numbers c or if f (resp. g) is proper on the level set g - 1 ( 0 ) (resp. f - 1 ( 0 ) ), then (f,g) is bijective.

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