Applications of degree estimate for subalgebras

Abstract

Let $K$ 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 $\varphi$ of $K<x,y>$ such that $\varphi (p(x,y))=x$ 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.

12 pages

References in corpus (1)

• Degree estimate for subalgebras

Cited by in corpus (1)

• Endomorphisms preserving coordinates of polynomial algebras