Denominators of rational polytopes in terms of hyperplane coefficients

Question

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 $m$?

Answer 1

Let $x$ be a vertex of $\mathcal{P}$. By Cramer's rule, there is an $n \times n$ matrix $C$ such that each coordinate of $x$ is an integer multiple of $\frac{1}{|\det(C)|}$, and the absolute value of each entry of $C$ is bounded by $m$. Therefore, by Hadamard's inequality, $$\kappa \leq m^n n^{n/2}.$$

The proof actually shows that $\kappa$ does not depend on $b$. Of course, the largest numerators appearing in $x$ will depend on $b$.