Given a non-constant polynomial $p\in\mathbb{Z}[x]=\alpha\prod_{k=1}^nx-\alpha_k$ how can I find the roots $R=\{\beta_1,\ldots,\beta_t\}\subseteq\{\alpha_1,\ldots,\alpha_n\}\subseteq\mathbb{C}$ with $|\beta_1|=\cdots=|\beta_t|$ and $|\beta_j|\ge|\alpha_k|$ for all $1\le j\le t$ and $\alpha_k\in\{\alpha_1,\ldots,\alpha_n\}\setminus R$?
The practical approach is to compute an approximation to the roots of the polynomial (with Schönhage, Aberth, Durand-Kerner, Graeffe, etc.), finding the maximum absolute value $M$, and taking all roots $\beta$ with $|\beta|>M(1-\varepsilon)$. But I'd like to have some guarantee that these rounding errors haven't caused me to miss a root or (more likely) include one wrongly. Perhaps it can be salvaged through careful numerical analysis, though, giving a proof that a given $\epsilon$ suffices?