Condition and algorithm for Decomposition of formal power series

$$F(x)= \Sigma_0^{\infty} a_i x^i$$ is formal power series, $a_i\in N\bigcup 0$,N is the set of natural number,under what condition may it be decomposed into a system of equations terms of which are polynomials of multivariables(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(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 F(x) a algebraic function,or a transcendental function?

Please do not downvote it if it is not very clear

edited Jan 27 at 10:52

asked Jan 27 at 5:16

XL
379●1●7

	Actually it is not very clear here and there. What are "N" and " 0 " ? What does "decompose a formal serie into a system of equations" mean? Where are te "polynomial of multivariable" What does the equation "F=FBCx + BCx"? What are B and C? Who is an algebraic or trascendental function? – Pietro Majer Jan 27 at 9:51
	@Pietro,Thank you for your comment,I have edited it again – XL Jan 27 at 11:15
	You should choose tags which fit to the question. – Martin Brandenburg Jan 27 at 11:46
	@Martin,Thank you ,but I do not know exactly what tags it is fit to.Maybe it is fit to abstract algebra,But I am not sure,somehow,I think it is related to Ideal and Ring of Algebra,and It may be regarded or investigated in the view of algebraic geometry. – XL Jan 27 at 13:36

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