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.