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2 Zhao W.

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Some properties of and open problems on Hessian nilpotent polynomials

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Zhao W.

nr 2

135-162

In the recent work [BE1], [Me], [Burgers] and [HNP], the well-known Jacobian conjecture ([BCW], [E]) has been reduced to a problem on HN (Hessian nilpotent) polynomials (the polynomials whose Hessian matrix is nilpotent) and their (deformed) inversion pairs. In this paper, we prove several results on HN polynomials, their (deformed) inversion pairs as well as on the associated symmetric polynomial or formal maps. We also propose some open problems for further study.

A generalization of Mathieu subspaces to modules of associative algebras

100%

Zhao W.

tom 8

nr 6

1132-1155

We first propose a generalization of the notion of Mathieu subspaces of associative algebras $$ \mathcal{A} $$, which was introduced recently in [Zhao W., Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra, 2010, 214(7), 1200–1216] and [Zhao W., Mathieu subspaces of associative algebras], to $$ \mathcal{A} $$-modules $$ \mathcal{M} $$. The newly introduced notion in a certain sense also generalizes the notion of submodules. Related with this new notion, we also introduce the sets σ(N) and τ(N) of stable elements and quasi-stable elements, respectively, for all R-subspaces N of $$ \mathcal{A} $$-modules $$ \mathcal{M} $$, where R is the base ring of $$ \mathcal{A} $$. We then prove some general properties of the sets σ(N) and τ(N). Furthermore, examples from certain modules of the quasi-stable algebras [Zhao W., Mathieu subspaces of associative algebras], matrix algebras over fields and polynomial algebras are also studied.