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]).

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

Homework #5 correction

There was a slight problem with question #2. Here is the corrected version:

2. a) Using a computer algebra system, show that the graph with edges
{(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,7),(3,4),(3,6),(3,9),(5,6),(6,8),(6,9),(7,8),(8,9)}

is 3-colorable. Use the Grobner basis you find to give an explicit 3-coloring. How many colorings are there?
b) Show that if you add edge (1,3) to the graph, it is still 3-colorable, and that the coloring is now unique up to permutation of the colors.

Homework 5 due date

I said in class, yet forgot to post on the blog, that homework #5 is due on April 22nd. By the way, I think I will assign one more homework on the 22nd that will be due May 4th. The final will also be assigned on May 4th, and will be due on the day the final is scheduled to be for our class, which is Thursday May 13th.

Also, here is the Macaulay2 file that illustrates the automated geometric proving and colorability of graphs applications from class.

I have also placed the book An Introduction to Grobner Bases by Adams and Loustaunau on reserve in the math library. The material on colorability of graphs, as well as the applications to linear programming are covered in chapter 2 of that book.

Homework #5

Problems from the text:

4.5: 3,8,11,12
4.6: 9
6.4: 3,8

Additional problems:

1. Using the Grobner basis technique from class (and of course, a computer algebra system), prove that Euler's nine point circle theorem is true. You may use either the fraction field method, or the saturation method to do this problem, but since the conclusions do not follow strictly from the hypotheses, you must use one or the other. A nice picture and discussion of the theorem is available here.

The statement is the following. Let ABC be a triangle. Then the midpoints of the sides, the feet of the altitudes, and the midpoints of the line segments joining the orthocenter (which is the intersection of the altitudes) and the vertices of the triangle, all lie on a single circle. The circle theorem of Appolonius is Euler's theorem in the case of a right triangle.

When I worked this problem out, I had 3 'u' variables, and 20 'x' variables, and therefore 20 'hypotheses'. There should be six 'conclusions' that you have to check. One way to proceed would be to consider the circle that goes through the midpoints of the sides, and show that the other 6 points are on this circle.

2. a) Using a computer algebra system, show that the graph with edges {(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,7),(3,6),(3,9),(5,6),(6,8),(7,8),(8,9)} is 3-colorable. Use the Grobner basis you find to give an explicit 3-coloring.
b) Show that if you add a new edge {(1,3)} to the graph, it is still 3-colorable, and that the coloring is now unique up to permutation of the colors.

3) Use the method of linear programming we learned in class to solve the following system of equations in the nonnegative integers:
3s_1+2s_2+s_3+2s_4=10
4s_1+3s_2+s_3=11
2s_1+4s_2+2s_3+s_4=10

Homework #4

3.3: 11a, 13, 14
4.1: 1, 4
4.3: 11, 12
4.4: 2,9

This homework is due on April 8th.

Homework #3 Comment

Please take a look at the comment on the homework #3 post.

Homework #3

Problems from the book:

2.6: 1, 4, 12
2.7: 2, 7
2.8: 8

1) Let I be a monomial ideal and > any term order. Show that any generating set for I is also a Grobner basis.

HW Confusion

In my original homework post, I said that only 15a of section 1.5 was required for the homework. Later, I posted how one could go about doing problem 15b online via Sage, making mention that it was in fact an assigned homework problem. Just to clarify, problem 15b will *not* be required of homework #2.

Sorry for any confusion.

Frank

HW Extension

It has come to my attention that the monomial ideals problem that I assigned is much easier with one more lemma under your belt. So, what I propose is that I extend the homework deadline to Thursday of this week, and add the lemma as an exercise that makes the rest of your homework much easier. It will also be useful at various points in the course.

Lemma:
Let I be an ideal of k[x_1,...,x_n]. Suppose that I has the property that for any f in I, every term of f also is in I. Show that I is a monomial ideal.

One can view this as the converse to Lemma 3 in section 2.4 of IVA; it is assigned as problem 1 in the exercises in that section. In some texts, this is the definition of a monomial ideal.

So, to recap, homework is now due on Thursday Feb 25th, and proving the preceding lemma is added to the list of assigned problems. This lemma should be extremely useful in the problem on sums, intersections, and radicals of monomial ideals, especially the radical problem. One can fumble through the sums and intersections problems without it, but trying the radical problem without it requires significantly more mathematical stamina.

Computer Algebra System

This is just a quick tutorial on how to use Sage Notebook to answer the 'computational' problem on this homework, namely problem 15.b in section 1.5. One does not need to use a computer to do this problem (it is not *that* hard to find compute the GCD), but for those of you that want to use a computer, here is how it can be done.

Go to www.sagenb.com and create a log in. You will be presented with the (currently empty) list of worksheets you have created. Create a new worksheet by clicking on the "New Worksheet" button along the top row. We will work through an example; note that this is *not* the example that you are asked to compute in the problem.

Near the top, you will see a drop down box that says 'sage'. Select 'macaulay2' from the dropdown box. After you type in each cell, be sure to click 'evaluate' at the bottom of the cell to see what the result of running each command will be.

In the first cell, type in the following:

R = QQ[x]
f = x^9 - 29*x^8 + 371*x^7 - 2747*x^6 + 12968*x^5 - 40460*x^4 + 83392*x^3 - 109440*x^2 + 82944*x - 27648

This tells sage that we are working in the ring QQ[x], and we are going to try to find the squarefree part of f. In the next block, put

fprime = diff(x,f)

This is just the derivative of f with respect to x. In the next box, put

gcd(f,fprime)

And then finally, in the next box put.

h = f // gcd(f,fprime)

To verify our work, check that both elements factor to give the right thing, so in two boxes, put

factor f

and

factor h

This is exactly what the problem told us would happen :)

Homework #2

Here is the next homework assignment:

By the way, below I use (f_1,...,f_s) to denote the ideal generated by the {f_i} instead of the angle brackets, since HTML doesn't like it when I try to use angle brackets.

1.5: 12, 13 (use linearity of the formal derivative, and prove this for monomials first!), 14, 15a
2.2: 2
2.3: 5

1. Let I and J be ideals of k[x_1,...,x_n]. Show that I intersect J, and I+J are ideals, where I+J = {f + g | f is in I and g is in J}.

2. Let I be an ideal of k[x_1,...x_n]. We say I is a monomial ideal if I can be generated by monomials m_1,...,m_s in the variables x_1,...,x_n. For example, (x^2) is a monomial ideal of k[x], while (x+x^2) is not.

a) Show that intersections and sums of monomial ideals are monomial.
b) Let I = (m_1,...,m_s). Compute a generating set for the radical of I, and show that it is also a monomial ideal.
c) Show that a monomial ideal I is radical if and only if I has a monomial generating set where each monomial is squarefree, i.e., the power on each variable in the monomial is 1. (i.e., xyz^2 is not squarefree, but xyz is).

The due date is February 23rd.

Frank

Homework #1

The first homework will be the following problems from Cox, Little, and O'Shea:

1.1: #5
1.2: #6,7,8
1.3: #6,8
1.4: #6

Also:

Find a Lagrange multiplier problem in your old calculus book (you still have one don't you ;) that has some messy equations that one must solve to get an answer, and solve it (how messy is up to your own discretion). We'll learn a much easier way of solving these systems later, and I will use some of these examples in class later on.

The due date for this assignment is February 11th.