Geometric condition of formal grammars solvability

In this paper, we continue the development of a method for studying formal grammars, which means systems of non-commutative polynomial equations. Such systems are solved in the form of formal power series (FPS) that represent non-terminal alphabet characters through terminal alphabet characters; the first component of the solution is a formal language. The method developed by the authors is based on the study of the commutative image of grammar and formal language. Namely, every FPS is associated with its commutative image, which is obtained if we assume that all symbols are commutative variables. A theorem that gives a sufficient geometric condition for the formal grammar to have a unique solution in the form of FPS is obtained: if the commutative images of non-commutative equations of a system define smooth complex analytical hypersurfaces at the point 0, and the normals to them drawn from this point are linearly independent, then the system of non-commutative equations has a unique solution in the form of FPS.

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