  We demonstrate that $C^{2,\alpha}$ estimates for the Monge-Amp\`{e}re
equation depend in a highly nonlinear way both on the $C^{\alpha}$ norm of the
right-hand side and $1/\alpha$. First, we show that if a solution is strictly
convex, then the $C^{2,\alpha}$ norm of the solution depends polynomially on
the $C^{\alpha}$ norm of the right-hand side. Second, we show that the
$C^{2,\alpha}$ norm of the solution is controlled by
$\exp((C/\alpha)\log(1/\alpha))$ as $\alpha \to 0$. Finally, we construct a
family of solutions in two dimensions to show the sharpness of our results.
