Infinite series representation for root of polynomials?

Question

Given a polynomial $p(x)=a_nx^n+\dots+a_1x+a_0$, can every root of the polynomial be represented as $\sum_{k=0}^\infty b_k$ with the $b_k$'s being a function of $a_0,\dots,a_n$ using only elementary operations of arithmetic and taking roots?

Answer 1

I think this is true at least formally if you allow the $b_i$ to have coefficients in $\overline{\mathbb{Q}}$. This is because, if $K$ is an algebraically closed field of characteristic $0$, then the field of Puiseux series with coefficients in $K$ is also algebraically closed, and by iterating this construction for each coefficient $a_i$ I think we get the desired result abstractly, although I am not sure what one can say about actual (as opposed to formal) convergence.

Answer 2

The difficult part is to get a good a priori estimate $\Omega\subset{\mathbb C}$ of the set $S$ of roots. Starting with any $z_0\in\Omega$, e.g., with rational coordinates, Newton's rule $$z_{n+1}:=z_n-{p(z_n)\over p'(z_n)}\qquad(n\geq 0)\ ,$$ i.e., $$b_0=z_0, \qquad b_{n+1}:=-{p(z_n)\over p'(z_n)}\qquad(n\geq 0),$$ provides a series $\sum_{k\geq 0} b_k$ converging to a point $\zeta\in S$ where the $b_k$ depend rationally on the coefficients of $p$ (and the chosen point $z_0$).

There is a famous paper by Smale on this: "The fundamental theorem of algebra and complexity theory", Bulletin of the American Mathematical Society ${\bf 4}$ (1981), 1–36.

Answer 3

Isn't this theorem related? As I understand the question, the OP asks wether it can be solvable by radicals, and Abel's theorem states that for polynomial equations of degree $n \geq 5$ there is no general solution.