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arXiv · 1011.6551

Applications of degree estimate for subalgebras

Yun-Chang Li·Jie-Tai Yu
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Abstract

Let $K$ be a field of positive characteristic and $K $ be the free algebra of rank two over $K$. Based on the degree estimate done by Y.-C. Li and J.-T. Yu, we extend the results of S.J. Gong and J.T. Yu's results: (1) An element $p(x,y)\in K $ is a test element if and only if $p(x,y)$ does not belong to any proper retract of $K $; (2) Every endomorphism preserving the automorphic orbit of a nonconstant element of $K $ is an automorphism; (3) If there exists some injective endomorphism $ϕ$ of $K $ such that $ϕ(p(x,y))=x$ where $p(x,y)\in K $, then $p(x,y)$ is a coordinate. And we reprove that all the automorphisms of $K $ are tame. Moreover, we also give counterexamples for two conjectures established by Leonid Makar-Limanov, V. Drensky and J.-T. Yu in the positive characteristic case.

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Yun-Chang Li, Jie-Tai Yu. 2010-11-30. Applications of degree estimate for subalgebras. https://arxiv.org/abs/1011.6551

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