Lower bound on the value of a polynomial

Question

I am interested in the following algorithmic question: Given as input an univariate polynomial $p(x) \in \mathbb{Q}(x)$ of degree at most $d$ as a list of coefficients, and $\epsilon \in \mathbb{Q}$, I want to test if $|p(\alpha)| \geq \epsilon$ where $\alpha$ is an algebraic number given by its minimial polynomial which has degree at most $d'$ (Say $d' > d$). Note that if there was a lower bound for $p(\alpha)$ as a function of the height $H$ and degreee $d$ of $p(x)$ and height $H'$ and degree $d'$ of $\alpha$, this task would be equivalent to evaluating $p(\alpha)$ to a precision dictated by a function $\epsilon(H, H', d, d')$ and we would be done. So equivalently, I'm asking for a lower bound of the form: "If $p(\alpha) \neq 0$ (which we know), then $p(\alpha) \geq \epsilon(H, H', d, d')$". Is such a lower bound easy to derive from the given information?