Let P be the set of polynomials {p(d)} in one variable, d, such that p(d) is an integer for all integer d. Of course this includes Z[d], but is larger, e.g. p(d) = d(d-1)/2. In particular, this includes Hilbert polynomials.
1. For each k, show c_k(d) := d(d-1)...(d-k+1)/k! is in P.
2. Show that the {c_k} are a basis of C[d].
3. Define Delta : P -> P by (Delta p)(d) = p(d) - p(d+1). Compute Delta(c_k).
4. Show that the {c_k} are a Z-basis of P, when we expand an element of P in {c_k}, the coefficients are in Z.
5. Define the _Hilbert series_ H_I(t) of a homogeneous ideal I as the series \sum_d h_I(d) t^d. Show that if I is the 0 ideal in C[x_0,...,x_n], then H_I(t) = 1/(1-t)^{n+1}.
6. Recall that r is a _zero divisor_ in R/I if r is not zero, and there exists s not zero such that rs = 0. Assume that I is graded and r is homogeneous of degree k and not a zero divisor.
It turns out that H_{I + < r >
7. Assume that I is graded, r is homogeneous of degree k, and the formula you figured out in #6 holds. Show that, conversely, this implies r is not a zero divisor.
