Let $\mathcal{P}$ be a convex polytope in $\mathbb{R}^n$ given in the form $\mathcal{P} = \{ x \in \mathbb{R}^n\colon A x\leq b \}$. Suppose that the entries of $A$ and $b$ are integers. Then it is well-known that $\mathcal{P}$ is a *rational* polytope, i.e., its vertices have rational (but not necessarily integral) coordinates. We might be interested in how far $\mathcal{P}$ is from being a lattice polytope. So let $\kappa$ be the biggest denominator appearing in any coordinate of a vertex of $\mathcal{P}$.

Finally, suppose further that the entries of $A$ and $b$ are bounded in absolute value by some $m$. Since there are only finitely many possible $A$ and $b$ in this case, we must be able to bound $\kappa$ in terms of $n$ and $m$. But I have no clue how $\kappa$ behaves. Which leads to my question.

**Question**: In this situation, what is an upper bound for $\kappa$ in terms of $n$ and $m$?