# Denominators of rational polytopes in terms of hyperplane coefficients

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 $$m$$?

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 $$b$$.