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28 March 2024
 
  » arxiv » 1011.6551

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Applications of degree estimate for subalgebras
Yun-Chang Li ; Jie-Tai Yu ;
Date 30 Nov 2010
AbstractLet $K$ be a field of positive characteristic and $K<x, y>$ 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 $p(x,y)in K<x,y>$ is a test element if and only if $p(x,y)$ does not belong to any proper retract of $K<x,y>$; (2) Every endomorphism preserving the automorphic orbit of a nonconstant element of $K<x,y>$ is an automorphism; (3) If there exists some injective endomorphism $phi$ of $K<x,y>$ such that $phi(p(x,y))=x$ where $p(x,y)in K<x,y>$, then $p(x,y)$ is a coordinate. And we reprove that all the automorphisms of $K<x,y>$ 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.
Source arXiv, 1011.6551
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