  In addition to the superconformal symmetry of the underlying Lagrangian, the
scattering amplitudes in planar N=4 super-Yang-Mills theory exhibit a new, dual
superconformal symmetry. We address the question of how powerful these
symmetries are to completely determine the scattering amplitudes. We use the
example of the NMHV superamplitudes to show that the combined action of
conventional and dual superconformal symmetries is not sufficient to fix all
the freedom in the tree-level amplitudes. We argue that the additional
information needed comes from the study of the analytic properties of the
amplitudes. The requirement of absence of spurious singularities, together with
the correct multi-particle singular behavior, determines the unique linear
combination of superinvariants corresponding to the n-particle NMHV
superamplitude. The same result can be obtained recursively, by relating the n-
and (n-1)-particle amplitudes in the singular collinear limit. We also
formulate constraints on the loop corrections to the superamplitudes, following
from the analytic behavior in the above limits. We then show that the
holomorphic anomaly of the tree amplitudes leads to the breakdown of dual
Poincare supersymmetry (which is equivalent to ordinary special conformal
supersymmetry) of the ratio of the NMHV and MHV superamplitudes at one-loop
level, but this anomaly does not affect dual conformal symmetry.
