Let us fix positive integers $m$ and $k$ and for $i\in\{0,\dots,k-1\}$ chose integers $a_i$, such that $|a_i|\le m$.
Now we form the polynomial $P(x)=x^k+\sum_{i=0}^{k-1} a_i x^i$.
From the rational root theorem we know that all rational solutions of $P$ are integers.
If we choose $P$ uniformly at random, what is the probability that it has an integer root? Is anything known about that?
Is it known how this probability changes as $m$ and $k$ changes?