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) ∈ 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 φ of K 〈x,y〉 such that φ (p(x,y)) = x where p(x,y) ∈ 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.