Geodesic
On the solution of a general algebraic equation by power series and applications in the theory of formal grammars
Prikladnaâ diskretnaâ matematika, no. 2 (2023), pp. 106-113

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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.
Keywords: general algebraic equation, power series, commutative image, formal language.
Mots-clés : Laurent series, polynomial grammar 
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     title = {On the solution of a general algebraic equation by power series and applications in the theory of formal grammars},
     journal = {Prikladna\^a diskretna\^a matematika},
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     year = {2023},
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O. I. Egorushkin; I. V. Kolbasina; K. V. Safonov. On the solution of a general algebraic equation by power series and applications in the theory of formal grammars. Prikladnaâ diskretnaâ matematika, no. 2 (2023), pp. 106-113. http://geodesic.mathdoc.fr/item/PDM_2023_2_a8/