  By taking the high-energy limit of the two-loop and the three-loop four-point
amplitudes in the maximally supersymmetric N=4 Yang-Mill theory (MSYM), we test
the validity of the loop expansion of the high-energy amplitude, beyond the
next-to-leading-logarithmic (NLL) accuracy. We compute the three-loop Regge
trajectory, and the two-loop and three-loop coefficient functions. These
quantities are relevant for the BFKL evolution beyond NLL, as well as building
MSYM two-loop and three-loop amplitudes with many legs in the high-energy
limit, which in turn may be used as a powerful check of the evaluation of the
corresponding exact amplitudes.
