Fix positive integers $k,n$, and let $p_m$ be the polynomial given by $$ p_m(x) = \sum_{k=0}^n m^{n-k} a_k x^k = x^n + ma_{n-1}x^{n-1} + m^2 a_{n-2} x^{n-2} + \dotsc + m^{n-1} a_1 x + m^n a_0, $$ where $m$ is a positive integer, and $a_0, \dotsc, a_n$ are positive real numbers.
Is it true that there exists $\epsilon > 0$ such that for all sufficiently large $m$ we have $$ \text{if $x_0$ and $x_1$ are distinct complex roots of $p_m$, then $|x_0 - x_1| > \epsilon$}? $$ In other words, I would like to obtain a uniform (in $m$) lower bound for the minimum distance between roots of $p_m$'s.