Representation of real or complex numbers as vector of coefficients of polynomials.

Question

Assuming we have a polynomial $$p(x) = \sum_{k=0}^N c_k x^k, \hspace{1cm} c_k \in \mathbb{N}$$

As far as I know we can create find polynomials having roots being any algebraic real number. Are there any standards of how to store such information?

For example:

$$p(x) = x^2-2=0 \Leftrightarrow \cases{x_1=-\sqrt{2}\\x_2 = \sqrt{2}}$$

If we store $$[c_2,c_1,c_0] = [1,0,-2]$$ in a vector that requires three integers and can store that we mean the real number $\sqrt{2}$. Something that no resolution of a fraction or floating point standard could. But how to store which of the roots we intend?

Answer 1

Using Thom codes: you can distinguish a root $x_0$ of $p$ between the other roots of $p$ by the signs of the numbers $p'(x_0),p''(x_0),\dots,p^{(N)}(x_0)$.

Relevant link: Manipulation of real roots of polynomials: Isolating Intervals or Thom's Codes.