1. For I a monomial ideal in R = C[x_1,...,x_n], let the multigraded Hilbert series be the sum of all the monomials in R that aren't in I.
a) Compute this for R = C[x,y] and I = 0, as a ratio of polynomials.
b) Generalize our inclusion-exclusion formula for Hilbert series to multigraded Hilbert series.
2. Let f,g be two polynomials, and fix a term order <. Assume that gcd(init(f),init(g)) = 1. Show that the reduction algorithm reduces the S-polynomial of f and g to zero (using f and g alone). (Moral: one needn't bother fiddling with the S-polynomial of f and g in this case.)
3. If f and g are monomials, what is their S-polynomial?
4. Compute a Gröbner basis for I = < x^2 - y, x^3 - z > with respect to lex order.
Can any of the generators in your list be safely left out?
5. Let p = x^2 + 3xy - y^2. Find a term order that makes x^2 the leading term, another that makes -y^2 the leading term, and show that 3xy isn't be the leading term under any term order.
(Oops; I left out the 3 in p the first time. That wasn't the point of the problem.)
