  We extend the analysis of Higgs-regulated planar amplitudes of N=4
supersymmetric Yang-Mills theory to four loops for the four-gluon amplitude and
to two loops for the five-gluon amplitude. Our calculations are consistent with
a proposed all-loop ansatz for planar MHV n-gluon amplitudes that is the analog
of the BDS ansatz in dimensional regularization. In all cases considered, we
have verified that the IR-finite parts of the logarithm of the amplitudes have
the same dependence on kinematic variables as the corresponding functions in
dimensionally-regulated amplitudes (up to overall additive constants, which we
determine). We also study various Regge limits of N=4 SYM planar n-gluon
amplitudes. Euclidean Regge limits of Higgs-regulated n \geq 4 amplitudes yield
results similar in form to those found using dimensional regularization, but
with different expressions for the gluon trajectory and Regge vertices
resulting from the different regulator scheme. We also show that the Regge
limit of the four-gluon amplitude is dominated at next-to-leading-log order by
vertical ladder diagrams together with the class of vertical ladder diagrams
with a single H-shaped insertion.
