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 :)
Friday, February 19, 2010
Friday, February 5, 2010
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
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.
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.
Tuesday, February 2, 2010
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.
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.
Thursday, January 28, 2010
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
See you on Tuesday,
Frank
Wednesday, January 27, 2010
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.
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.
Tuesday, January 26, 2010
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
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
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