Let J = I + < r >.
Show that
a) For each n, h_J(n) is at least h_I(n) - h_I(n-k).
b) If they are equal for all n, then r is not a zero divisor.
2. A list {r_1, r_2, ..., r_m} is called a regular sequence if each r_j is not a zero divisor in R/< r_1, ..., r_{j-1} >.
If {b,c} is a regular sequence, show that {c,b} is a regular sequence.
Oops: I had meant b,c to be homogeneous. (It's true even if they're not, but don't bother with that.)
3. Let p(n) be a polynomial of degree d, and k a number. Show that q(n) = p(n) - p(n-k) is a polynomial of degree d-1.
4. For p(n) a polynomial, let Delta p be the polynomial with values (Delta p)(n) = p(n) - p(n-1).
Notice that if p only takes integer values (when fed integers), then Delta p does so too.
a) Show that Delta {n choose k} = {
b) Show that every polynomial p(n) is a linear combination of the polynomials {n choose k}, where k goes from 0 to degree p.
c) Give an example of a polynomial with noninteger coefficients that nonetheless always produces integers.
d) Show that every integer-valued polynomial p(n) is a linear combination with integer coefficients of the polynomials {n choose k}, where k goes from 0 to degree p.
(In particular, this applies to Hilbert polynomials.)
5. Let I in C[x_1...x_5] be generated by {x_i x_j - x_k x_l}, for all i,j,k,l such that i+j=k+l.
a) Show this is a Gr\"obner basis with respect to lex order.
b) Find the reduced Gr\"obner basis.
c) Decompose V(the initial ideal).
d) Compute the Hilbert polynomial.
