  We derive useful reduction formulae which express one-loop Feynman integrals
with a large number of external momenta in terms of lower-point integrals
carrying easily derivable kinematic coefficients which are symmetric in the
external momenta. These formulae apply for integrals with at least two more
external legs than the dimension of the external momenta, and are presented in
terms of two possible bases: one composed of a subset of descendant integrals
with one fewer external legs, the other composed of the complete set of
minimally-descendant integrals with just one more leg than the dimension of
external momenta. In 3+1 dimensions, particularly compact representations of
kinematic invariants can be computed, which easily lend themselves to
spinor-helicity or trace representations. The reduction formulae have a close
relationship with D-dimensional unitarity cuts, and thus provide a path towards
computing full (all-epsilon) expressions for scattering at arbitrary
multiplicity.
