Let I = < f_1, ..., f_k > where the f_i are each homogeneous polynomials.
1. Show that I has a Grobner basis consisting of homogeneous polynomials.
2. Assume hereafter that the {f_i} are a reduced Grobner basis.
Recall that the Hilbert function h_I(n) is the dimension of R_n / I_n.
Call a monomial standard if it isn't divisible by any of the leading monomials of the {f_i}.
Show that h_I(n) is the number of standard monomials of degree n.
3. Show that h_I = h_{init I}, so H_I = H_{init I} (the Hilbert series).
4. The polynomial b^2 - ac has two possible leading terms (depending on term order), so I = < b^2 - ac > has two possible init I. Compute each of their Hilbert series (and show they are equal, as problem 3 predicts).
5. Let I = < the entries of M^2, where M is a 2x2 matrix >. Compute the Hilbert series H_I.
6. Let J be the larger ideal containing I and also the generator Trace(M). Compute the Hilbert series H_J.
