  We investigate conditions for "simultaneous normalizability" of a family of
reduced schemes, i.e., the normalization of the total space normalizes, fiber
by fiber, each member of the family. The main result (under more general
conditions) is that a flat family of reduced equidimensional projective complex
varieties X_y with parameter y ranging over a normal space--algebraic or
analytic--admits a simultaneous normalization if and only if the Hilbert
polynomial of the integral closure of the structure sheaf O_{X_y} is locally
independent of y. When the X_y are curves projectivity is not needed, and the
statement reduces to the well known \delta-constant criterion of Teissier.
Proofs are basically algebraic, analytic results being related via standard
techniques (Stein compacta, etc.) to more abstract algebraic ones.
