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 $a\_0, \dots, a\_n$?



How close to an integer can a polynomial root be?
A bound follows from the general root separation theory. See [Schonhage's 2006 paper](http://ac.els-cdn.com/S0747717106000472/1-s2.0-S0747717106000472-main.pdf?_tid=3e55d828-be9e-11e4-8390-00000aab0f26&acdnat=1425054938_40447d42a9f0f57fb2aec11709601c4c) (Journal of Symbolic Computation)> (see inequality (3) and the discussion following, about replacing the discriminant by $1.$)


