On a solution of polynomial grammars and the general algebraic equation

In this paper, we investigate the solvability of formal grammars, by which we mean systems of non-commutative polynomial equations, in the case of one equation. Formal grammars are solved in the form of formal power series (FPS), which express nonterminal symbols of the language through terminal symbols; the first component of the solution is the formal language. The authors develop a method based on the study of the commutative image of grammar and language, which is obtained if in any FPS the symbols of the alphabet are considered commutative variables. A theorem is obtained that gives a power series expansion of the solution to a general algebraic equation, and also allows us to investigate the solvability in the form of an FPS of a polynomial grammar consisting of one equation.