  The Hilbert-Samuel function and the multiplicity function are fundamental
locally defined invariants on Noetherian schemes. They have been playing an
important role in desingularization for many years. Bennett studied upper
semicontinuity of the Hilbert-Samuel function on schemes and proved that it is
non increasing under permissible blowing ups. The latter are blowing ups at
regular subschemes along which the singular scheme is normally flat. For a
reduced scheme, the Hilbert-Samuel function is constant if and only if it is
regular: this translates the question of resolution of singularities into a
problem of lowering the Hilbert-Samuel function. We show here that this result
can be extended to non reduced schemes as follows: Given a locally Noetherian
scheme X such that the local rings are excellent for every point, then the
Hilbert-Samuel function is constant on X if and only if X is normally flat
along its reduction and the reduction itself is regular.
