arXiv subjects
Publications and source records attributed to Yun-Chang Li.
Let $K$ be a field of characteristic zero, $K[x,y]$ be the polynomial ring in two variables. Let $ϕ=(f, g)$ be an endomorphism of $K[x,y]$. It is proved that if $ϕ$ maps each coordinate to a generator of some proper retract, then it is an automorphism. As a corollary, the retract preserving problem is solved for both polynomial ring over $K$ and free algebra over an arbitrary field when $n=2$.
It is proved that the Jacobian of a k-endomorphism of k[x_1,...,x_n] over a field k of characteristic zero taking every tame coordinate to a coordinate, must be a nonzero constant in k. It is also proved that the Jacobian of an R-endomorphism of A:=R[x_1,...,x_n] (where R is a polynomial ring in finite number of variables over an infinite field k), taking every R-linear coordinate of A to an R-coordinate of A, is a nonzero constant in k.
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.