Abstract. We consider absolutely free nonassociative algebras and, more generally, absolutely free algebras with (maybe infinitely) many multilinear operations. Such algebras are described in terms of labeled reduced planar rooted trees. This allows to apply combinatorial techniques to study their Hilbert series and the asymptotics of their coefficients. These algebras satisfy the Nielsen-Schreier property and their subalgebras are also free. Then, over a field of characteristic 0, we investigate the subalgebras of invariants under the action of a linear group, their sets of free generators and their Hilbert series. It has turned out that, except in the trivial cases, the algebra of invariants is never finitely generated. In important partial cases the Hilbert series of the algebras of invariants and the generating functions of their sets of free generators are expressed in terms of elliptic integrals.

Abstract. We study z-automorphisms of the polynomial algebra K[x, y, z] and the free associative algebra K〈x, y, z 〉 over a field K, i.e., automorphisms which fix the variable z. We survey some recent results on such automorphisms and on the corresponding coordinates. For K〈x, y, z 〉 we include also results about the structure of the z-tame automorphisms and algorithms which recognize z-tame automorphisms and z-tame coordinates.

Abstract. The classical theorem of Weitzenböck states that the algebra of invariants K[X] g of a single unipotent transformation g ∈ GLm(K) acting on the polynomial algebra K[X] = K[x1,..., xm] over a field K of characteristic 0 is finitely generated. This algebra coincides with the algebra of constants K[X] δ of a linear locally nilpotent derivation δ of K[X]. Recently the author and C. K. Gupta have started the study of the algebra of invariants Fm(V) g where Fm(V) is the relatively free algebra of rank m in a variety V of associative algebras. They have shown that Fm(V) g is not finitely generated if V contains the algebra UT2(K) of 2 ×2 upper triangular matrices. The main result of the present paper is that the algebra Fm(V) g is finitely generated if and only if the variety V does not contain the algebra UT2(K). As a by-product of the proof we have established also the finite generation of the algebra of invariants T g nm where Tnm is the mixed trace algebra generated by m generic n × n matrices and the traces of their products.