Test Polynomials for Automorphisms of Polynomial and Free Associative Algebras

In this paper we consider test polynomials in the polynomial algebra and the free associative algebra. A test polynomial is defined by the following property: every endomorphism which fixes the polynomial is an automorphism. We construct families of test polynomials for the polynomial algebra and the free associative algebra and show how different techniques may be used in the investigation of test polynomials. We also introduce the notion of a test vector space and determine all test vector spaces of the free associative algebra.