Test elements, retracts and automorphic orbits

Let A2 be a free associative or polynomial algebra of rank two over a field K of characteristic zero. Based on the degree estimate of Makar-Limanov and J.-T. Yu, we prove: (1) An element p ∈ A2 is a test element if p does not belong to any proper retract of A2; (2) Every endomorphism preserving the automorphic orbit of a nonconstant element of A2 is an automorphism. © 2008 Elsevier Inc. All rights reserved.link_to_subscribed_fulltex