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Let F = X + H be a cubic homogeneous polynomial automorphism from to . Let be the nilpotence index of the Jacobian matrix JH. It was conjectured by Drużkowski and Rusek in [4] that . We show that the conjecture is true if n ≤ 4 and false if n ≥ 5.
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 (resp. ), then (f,g) is bijective.
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