An algebraicity criterion for the sum of a power series (a generalization of Kronecker's criterion) and its application

A characterization of the coefficients in the power series of an algebraic function could be obtained by generalizing Kronecker's rationality criterion. The algebraic function can be obtained by finding an adequate method for stating conditions on the coefficients in power series obtained as a result of substituting this series into the polynomial equation. The algebraicity of function, which is equivalent to the existence of a nontrivial solution to a system is also equivalent to the nontrivial solvability of the infinite system of the extended equations. A function holomorphic on some domain is proved to be algebraic if and only if it is algebraic in each variable at fixed values of the other variables. In the case of a context-free language, the function obtained is found to be holomorphic in a neighborhood of the origin and algebraic.