Here is an additional hint on 3b: Note that with such a substantial hint, I expect cleaner proofs ;)

i) Prove that (I:f) is monomial by induction on the number of terms of f, which we may assume are all *not* in I by part a). To do this, let g be in (I:f). We need to show that every term of g is in (I:f). Consider gf. It is in I, so LT(g)LT(f) is in I.

Now consider (I:LT(g)f). Now (I:f) is a subset of (I:LT(g)f). Also, note that (I:LT(g)f), if prime, is monomial by the inductive hypothesis (Why?!?). Now show that if (I:f) is a proper subset of (I:LT(g)f), then LT(g) \in (I:f), and now repeat the argument for g-LT(g).

The following facts will help you to prove that (I:f) = (I:m) for some monomial m:

ii) Note that since I is monomial, one has that (I:f) = intersection of (I:f_i) for all the terms f_i of f. [You should prove this]

iii) Recall a (jazzed up version of a) homework problem: If P is prime and the product of the ideals I_1, I_2, ..., I_t (denoted I_1I_2I_3...I_t) is in P, then I_j is in P for some j. You may just cite this version without giving the (easy) proof using induction and the previous homework problem.

To show that if I is squarefree, then one can take m to be squarefree, you should look at Theorem 4.4.11 in the book.

## Tuesday, May 11, 2010

## Saturday, May 8, 2010

### Office hours this weekend

Unfortunately, I won't be able to hold my regular office hours this week. To compensate, I'm holding office hours this weekend:

Saturday 3:30 to 5:30

I can also see people on Sunday morning if you send me an e-mail

Best of luck with your final exam.

Saúl

## Friday, May 7, 2010

### Final Exam

The final exam is available here.

Note that the wording on problem 2 is changed slightly. One must assume that f is homogeneous in part a, and that I is a homogeneous ideal in part b) [see problem 1 for a definition of homogeneous].

Note that the wording on problem 2 is changed slightly. One must assume that f is homogeneous in part a, and that I is a homogeneous ideal in part b) [see problem 1 for a definition of homogeneous].

## Sunday, May 2, 2010

### Office hours on 05/05/2010

My office hours for Wednesday May 5 will be from 1 PM to 2 PM in Malott 112.

Saúl

## Friday, April 30, 2010

### Homework #6 correction

I had the numbers wrong in the additional problem that was assigned. The correct problem is now listed in the homework (I changed the 'b' vector from [-323,1035,654] to [-323,389,8]).

## Monday, April 26, 2010

### Homework #6

Sorry for the delay in assigning the homework. Posting the problem set completely slipped my mind until Sunday evening. The due date for this homework will be May 6th to account for the delayed assignment time.

6.1: 4

6.2: 4

6.3: 5,7,10

7.1: 5,8

1. Find an optimal integer solution to the following integer programming problem using the cost function c(s_1,s_2,s_3,s_4,s_5) = 10s_1+100s_2+100s_3+10s_4+s_5, where

2s_1+5s_2-3s_3+s_4-2s_5=-323

s_1+7s_2+2s_3+3s_4+s_5=389

4s_1-2s_2-s_3-5s_4+3s_5=8

6.1: 4

6.2: 4

6.3: 5,7,10

7.1: 5,8

1. Find an optimal integer solution to the following integer programming problem using the cost function c(s_1,s_2,s_3,s_4,s_5) = 10s_1+100s_2+100s_3+10s_4+s_5, where

2s_1+5s_2-3s_3+s_4-2s_5=-323

s_1+7s_2+2s_3+3s_4+s_5=389

4s_1-2s_2-s_3-5s_4+3s_5=8

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