  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 $\phi$ 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.
