$$F(x)= \Sigma_0^{\infty} a_i x^i$$ 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
