Additional Hint on 3b)

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.

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

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].

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

Updated example file

The updated Macaulay2 example file from class is available here.

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

HW 5, what to hand in

I have received questions about what needs to be hand in for the computer problems. Here is what I think you need FOR ALL THREE PROBLEMS: The input (hypothesis, conclusions, edges, etc), the output, and a few sentences explaining what the output means and why it gives you (or not) what you want. It would be nice if you mark these clearly, so it is easier for me to find. Also, it is perfectly find to hand in printouts of what the input/output (no need to write them down with your own handwriting).

Good luck with your HW.

Saúl

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

Exam Solutions

Here are solutions to the exam.

Frank

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.

Exam and Macaulay2 file

Here is the Macaulay2 file alluded to in the exam.

Homework #3 Comment

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

Macaulay2 script

The Macaulay2 script solving the Lagrange multiplier problem can be found here.

One can perform this computation in a Sage Notebook screen with macaulay2 enabled as I mentioned in an earlier blog post, or you can try installing Macaulay2 and emacs on your personal machine by following the installation instructions on the Macaulay2 Home Page.

If you have any problems setting it up, then I can try and help you via email.

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 Comments

The fist HW set has been graded. In general people did pretty well. Here are a couple of comments to consider for future assignments.

1) Please be neat. Your paper should not have scribbles or crossed out words. It should be correctly stapled, and the writing should be legible.

2) Write in full sentences. Look at the proofs in your textbook, they are all written in full sentences, and not as a collection of equations.

If there are any complaints about the grading, don't hesitate to come to office hours to talk about it. My office hours are Monday and Wednesday from 1:30 to 2:30 PM in Malott 112.

Thank you,

Saul

Timetable up until Exam 1

Now that we are nearing the first exam time (already!) I thought I would let you know the timetable that I had in mind for the next assignment, as well as the first exam:

HW #2 due Feb 25th
HW #3 assigned Feb 25th
HW #3 due Mar 9th
Exam #1 assigned Mar 9th
Exam #1 due Mar 18th

Frank

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

Ideals and Varieties

In this past week, we have introduced a fair amount of new objects, so lets recap what we have done, as well as give an indication as to where we are headed.

We defined an affine variety V to be the set of points in affine n-space k^n satisfying a system of polynomial equations f_1 = ... = f_k = 0. There are many times in applications where one wants to understand properties of such a set. Varieties can describe many things, such as allowable positions of robotic arms, solutions to Sudoko puzzles, and others.

The description of an affine variety as a zero locus of a collection of polynomials is called an implicit description. When using such a description, it is easy to determine if a point in k^n is on the variety, but hard to find all the points. A description of (most of) the points on the variety via rational functions is called a parameterization of the variety. Parameterizations are nice because they make it easy to find points on the variety, but hard to know whether a given point is in the variety.

We did several examples of both polynomial and rational parameterizations of some different varieties, namely the circle, ellipse, parabola, and the tangent surface to the twisted cubic. One can always go from a rational parameterization of an affine variety to an implicit description using elimination theory, but not all varieties have a single rational parameterization of 'most' of the points. We'll see this later on in the class.

In linear algebra, one introduces the idea of a subspace in order to study solutions of linear systems. In order to study varieties, one introduces the notion of an ideal. An ideal is a subset of the polynomial ring that is closed under sums and "polynomial scalings", that is, if r is a polynomial, and f is in the ideal, then rf is also in the ideal. For us, we will be interested in the smallest ideal containing a set of polynomials f_1,...,f_k. This ideal can be thought of as the set of 'polynomial consequences' of the equations f_1 = 0, ..., f_k = 0. We'll spend a fair amount of time studying ideals, and algorithms that pertain to them, in order to study varieties.

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.

Cancelled class today

I am sorry, but my substitute Allen Knutson is sick today and is not able to cover my class for me while I am away. We will just have to work extra hard for the remainder of the semester :)

See you on Tuesday,

Frank

First day recap, and an important announcement

On 1/26 I spent some time motivating the problems we will look at by first reminding you of the types of things one does in a linear algebra class. Some basic problems that one encounters are:

Solving systems of linear equations
Structure of the solutions to a linear system
Detecting 'redundant' systems
Parametric vs. Implicit descriptions of objects

We will handle each of these problems in this class as well, but for polynomial equations rather than linear ones, with an emphasis on the algorithms that one uses to solve these problems.

One thing that I forgot to mention is that I am going to be out of town for class on Thursday 1/28. Class is still scheduled to meet, and
Allen Knutson will be covering class for that day. I assure you that you are in excellent hands.

Allen will begin to tell you about affine space, as well as some first examples of solution sets of polynomials on Thursday, and will perhaps begin to introduce you to ideals in a polynomial ring, which is the algebraic object that replaces the 'span' of a linear system of equations in our case.

Course Book, other announcements

Hello everyone!

It's a pleasure to be able to introduce you to the interesting world of computational algebra. We'll learn about solving polynomial systems of equations, and their applications to some (seemingly easy, yet difficult) robotics, colorability of graphs, and automated geometric theorem proving (which you had *that* in grade 9, right?). Along the way we will learn about some deep mathematics and discover that what we are talking about is only the tip of the iceberg.

The book that I will be following for the course is Cox, Little, and O'Shea's "Ideals, Varieties, and Algorithms" published by Springer-Verlag. The bookstore has it, but if you would also like an electronic copy, then as one of the privileges of being a Cornell undergraduate, your library has phenomenal access to online mathematics texts from Springer. If you access the below link while on Cornell's campus, you will gain access to the chapters of the book (sadly, there is not a 'download book' button).

Ideals, Varieties, and Algorithms

At various points in the semester, we will also be using an open source mathematics package called Sage, and a variant of it called Sage Notebook. Sage Notebook provides access to Sage via a Java interface on the web, allowing you to tinker with mathematics with ease. It also provides a convenient interface to a more advanced computer algebra package that we will use later on in the semester, Macaulay2.

The course grade will consist of homeworks (50%), a midterm (20%) and final (30%), with both exams being take-home exams. The homework will be due roughly once a week or two weeks, depending on the assignment. As of right now, the midterm is scheduled to be handed out 3/4 and due on 3/11 and the final is scheduled to be handed out 5/6 and due the scheduled day of the final, which is 5/13. This is subject to change.

Your TA for the course is Saul Blanco Rodriguez, and he has set tentative office hours for MW 1:30-2:30. My office hours are just by discovery - I am in my office from 9-5 most days and if you find me in my office (Malott 587), then by all means ask questions. If there is a particular time that you would like to speak with me, then please email me at myname(one word) at math dot cornell dot edu and we can set up a time.

See you soon,

Frank