  In this paper we investigate some algebraic and geometric consequences which
arise from an extremal bound on the Hilbert function of the general hyperplane
section of a variety (Green's Hyperplane Restriction Theorem). These geometric
consequences improve some results in this direction first given by Green and
extend others by Bigatti, Geramita, and Migliore.
  Other applications of our detailed investigation of how the Hilbert
polynomial is written as a sum of binomials, are to conditions that must be
satisfied by a polynomial if it is to be the Hilbert polynomial of a
non-degenerate integral subscheme of $\mathbb P^n$ (a problem posed by R.
Stanley). We also give some new restrictions on the Hilbert function of a zero
dimensional reduced scheme with the Uniform Position Property.
