Let $P(x)=a_nx^n+...+a_0$ be a polynomial where the coefficients satisfy $a_i \in \mathbb{Z}\cap [-m,m]$ for all $1\leq i \leq n$ and $m$ a positive number. Let $x_0$ be a (local) maximum of $P$ and assume that $P(x_0)>0$. Find an explicit lower bound $\varepsilon_{m,n}>0$ depending on $m$ and $n$ such that $$P(x_0)>\varepsilon_{m,n}.$$
In words: Assume you have incomplete information of a polynomial which has a positive maximum. How can one estimate the size of this maximum from below. To me this seems to be a natural question to ask but I haven't got the slightest clue on how to approach it.