A solvability condition for arbitrary formal grammars

In the paper, approaches to solving the systems of non-commutative polynomial equations in the form of formal power series (FPS) based on the connection with the corresponding commutative equations are developed. Every FPS is mapped to its commutative image — power series, which is obtained under the assumption that the symbols denote commutative variables assigned as values in the field of complex numbers. The consistency of the system of noncommutative polynomial equations, which is not directly connected with the consistency of its commutative image, is investigated. However, the analogue of implicit mapping theorem to arbitrary formal grammars (non-commutative systems) is obtained, namely if the rank of Jacoby matrix for the commutative image of a system of equations is maximal, then the initial noncommutative system of equations has a unique solution in the form of FPS.