On the solution of a general algebraic equation by power series and applications in the theory of formal grammars

A general algebraic equation is considered, and the problem is to find its solution using power series or Laurent series depending on the coefficients of the equation. A solution is obtained in the form of a Laurent series, the coefficients of which are expressed in terms of the coefficients by formulas in a “closed” form, when the number of terms in the formula does not increase with the number of the coefficient. In the applied aspect, a general algebraic equation is considered as a commutative image of the corresponding equation with non-commutative symbols, which, in turn, is interpreted in the theory of formal grammars as a polynomial grammar. It is shown that such a grammar does not generate a formal language (it does not have a solution in the form of a formal power series), since its commutative image has a solution in the form of a Laurent series containing negative degrees of variables, while division in the theory of formal grammars is not defined.