3 fixed grammar

$$F(x)= \Sigma_0^{\infty} a_i x^i$$ is formal power series, $a_i\in N\bigcap 0$,under N\bigcup 0$,N is the set of natural number,under what condition may it be decomposed into a system of equations term terms of which is polynomial are polynomials of multivariablesmultivariables(EDIT:decomposed means we can get F(x)expressed by x by solving the system of equations)? Decomposition is like: $$F(x)= \Sigma_1^{\infty} x^{3i}$$ may be decomposed into: F=FBCx+BCx B=x C=x F(x)=F(x)B(x)C(x)x+B(x)C(x)x B(x)=x C(x)=x If it may,is there algorithm to do that? these power series may be regarded as a complex function with convergence radius. Question: when is it F(x) a algebraic function,or a transcendental function? Please do not downvote it if it is not very clear        2 fixed grammar 1 Condition and algorithm for Decomposition of formal power series $$F(x)= \Sigma_0^{\infty} a_i x^i$$ is formal power series,$a_i\in N\bigcap 0\$,under what condition may it be decomposed into a system of equations term of which is polynomial of multivariables?

Decomposition is like: $$F(x)= \Sigma_1^{\infty} x^{3i}$$ may be decomposed into:

F=FBCx+BCx

B=x

C=x


If it may,is there algorithm to do that?

these power series may be regarded as a complex function with convergence radius. Question: when is it a algebraic function,or a transcendental function?

Please do not downvote it if it is not very clear