How close to an integer can a polynomial root be? Suppose I have a polynomial $p(x) = a_n x^n + ... + a_0$ where $a_n, \dots, a_0$ are integers. I would like to show that any root of this polynomial is either an integer or is far from an integer. That is, if $r$ denotes the fractional part of a root $x$, I want to show that either $r = 0$ or $r > w$ for some real valued $w$. 
If the polynomial $p$ has degree one this is easy --- I know either $w = 0$ or $w \geq \frac{1}{|a_1|}$. 
Is there a similar bound that can be shown for arbitrary polynomials, where the minimum non-zero value of $w$ can be bounded in terms of the sizes of $a_0, \dots, a_n$?
 A: This is a standard question in diophantine approximation.  See for example Chapter 3 of Waldschmidt's book Diophantine Approximation of Linear Algebraic Groups.  Here is the simplest bound that Waldschmidt gives.
Assuming $a_0\ne 0$, let $H := \max_{0\le i\le n} |a_i|$.  We will show that if $\alpha$ is any nonzero root of your polynomial, then $|\alpha| > 1/(H+1)$.
If $|\alpha| \ge 1$ then the inequality trivially holds.  Otherwise, if $|\alpha|<1$, then
$${1\over |\alpha|} \le \left|{a_0\over \alpha}\right| = \left| a_1 + a_2\alpha + \cdots + a_n \alpha^{n-1} \right| \le H(1+|\alpha| + \cdots+|\alpha|^{n-1}) < {H\over 1-|\alpha|},$$
which rearranges to $|\alpha| > 1/(H+1)$.

Edit: See David Lampert's comment below for how to use the above bound to answer the question that David Harris asked.
A: A bound follows from the general root separation theory. See Schonhage's 2006 paper (Journal of Symbolic Computation)> (see inequality (3) and the discussion following, about replacing the discriminant by $1.$)
