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?
[Math] Infinite series representation for root of polynomials
polynomialssequences-and-series
Best Answer
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.