Abstract
In the article, we prove that any polynomial endomorphism of , where is a field of characteristic zero, of the form ‘identity + homogeneous component of degree two’ is linearly triangularizable when the Jacobian matrix of the quadratic term is nilpotent of index at most three. In particular, this proves the well-known Meisters and Olech Conjecture. In our approach, besides methods of the classical matrix theory, some elements of the theory of commutative non-associative algebras appear in the context of the so-called polarization algebras associated with quadratic homogeneous endomorphisms. A powerful role of polarization algebras in the investigation of polynomial endomorphisms is revealed in a recent paper of Umirbaev. His results and comments were the main inspiration for us.
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