Automorphic equivalence problem for free associative algebras of rank two

Abstract. Let K〈x, y 〉 be the free associative algebra of rank 2 over an algebraically closed constructive field of any characteristic. We present an algorithm which decides whether or not two ele-ments inK〈x, y 〉 are equivalent under an automorphism ofK〈x, y〉. A modification of our algorithm solves the problem whether or not an element in K〈x, y 〉 is a semiinvariant of a nontrivial automor-phism. In particular, it determines whether or not the element has a nontrivial stabilizer in AutK〈x, y〉. An algorithm for equivalence of polynomials under automor-phisms of C[x, y] was presented by Wightwick. Another, much simpler algorithm for automorphic equivalence of two polynomi-als in K[x, y] for any algebraically closed constructive field K was given by Makar-Limanov, Shpilrain, and Yu. In our approach we combine an idea of the latter three authors with an idea from the unpubished thesis of Lane used to describe automorphisms which stabilize elements of K〈x, y〉. This also allows us to give a simple proof of the corresponding result for K[x, y] obtained by Makar-Limanov, Shpilrain, and Yu. 1