Applications of degree estimate for subalgebras

TL;DR

This paper extends degree estimate results in free algebras over fields of positive characteristic, providing new characterizations of test elements, automorphisms, and coordinates, and offering counterexamples to existing conjectures.

Contribution

It generalizes previous results by Li and Yu, establishing novel criteria for test elements, automorphisms, and coordinates in free algebras, and refutes some conjectures in positive characteristic.

Findings

Characterization of test elements via proper retracts

Automorphisms preserving automorphic orbits are automorphisms

Counterexamples to conjectures in positive characteristic

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 $\phi$ 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.