The test rank of a solvable product of free abelian Lie algebras

Let [Formula: see text] be the [Formula: see text]th solvable product of free abelian Lie algebras of finite rank. We prove that the test rank of [Formula: see text] is one less than the number of the factors. We also give a test set for endomorphisms of [Formula: see text].

On a solvable class of product-type systems of difference equations

It is shown that the following class of systems of difference equations zn+1 = ?zanwbn, wn+1 = ?wcnzdn-2, n ? N0, where a,b,c,d ? Z, ?, ?, z-2, z-1, z0,w0 ? C \ {0}, is solvable, continuing our investigation of classification of solvable product-type systems with two dependent variables. We present closed form formulas for solutions to the systems in all the cases. In the main case, when bd ? 0, a detailed investigation of the form of the solutions is presented in terms of the zeros of an associated polynomial whose coefficients depend on some of the parameters of the system.