On generators of polynomial algebras in two commuting or non-commuting variables

AbstractAn element of a free associative algebra A2 = K〈x1, x2〉 is called primitive if it is an automorphic image of x1. We address the problem of detecting primitive elements of A2: we present an algorithm that distinguishes primitive elements, and also give a couple of very handy necessary conditions for primitivity that allow one to rule out many sorts of non-primitive elements of A2 just by inspection. We also give a structural description of the automorphism groups Aut(A2) and Aut(P2) (where P2 = K[x1, x2] is the polynomial algebra in two variables over the same ground field K) which is different from previously known descriptions