 
Summary: Problem Set 10
For this entire problem set, R = k[x1, x2, . . . , xn] with k a eld.
Recall that if F, g1, g2, . . . , gt R, then there is an expression of the form:
G = F 
t
i=1
aigi where ai R and
1) None of the monomials of G are in (in>(g1), in>(g2), . . . , in>(gt)).
2) in>(F) in>(aigi) for every i.
Also recall that such a G is called a remainder upon dividing F by the elements
g1, g2, . . . , gt and that such remainders are not usually unique.
Problem 1. Show that if G = {g1, g2, . . . , gt} is a Gröbner basis and F R then
the remainder upon dividing F by the elements of G is unique.
Problem 2. Let I = (x3
y  x2
, x2
 xy). Compute I : x2
using the algorithm
for ideal quotients. (You can use Macaulay 2, just don't use the ideal quotient
command).
