Quasi-translations and counterexamples to the Homogeneous Dependence Problem

de Bondt, Michiel

doi:10.1090/S0002-9939-06-08335-3

Quasi-translations and counterexamples to the Homogeneous Dependence Problem
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Proc. Amer. Math. Soc. 134 (2006), 2849-2856 Request permission

Abstract:

In this article, the author gives counterexamples to the Linear Dependence Problem for Homogeneous Nilpotent Jacobians for dimension 5 and up. This problem has been formulated as a conjecture/problem by several authors in connection to the Jacobian conjecture. In dimension 10 and up, cubic counterexamples are given. In the construction of these counterexamples, the author makes use of so-called quasi-translations, a special type of invertible polynomial maps. Quasi-translations can also be seen as a special type of locally nilpotent derivations.

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