|Math 191: Algorithms for Elementary Algebraic Geometry
Time and Place: MWF 1-1:50pm, Dodd Hall 162
E-mail address: matthiasmath.ucla.edu
Office: Mathematical Sciences Building 5614
Office Phone: (310) 206-8576
Office Hours: M 2:30-4pm, W 2:30-4pm, or by appointment.
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Click here to download the course handout.
objects can often be described as the solution sets of algebraic
equations. Simple examples in three-dimensional space are curves
|(the set of triples (x,y,z) of real numbers satisfying the
equations y=x², z=x³)
and surfaces like
||(the set of triples (x,y,z) of real numbers solving the
In this course, we will investigate questions such as: How can one
compute the equations for the intersection or union of two such
objects? How can one determine whether two systems of algebraic
describe the same geometric object?
These are basic questions at the foundations of algebraic geometry
This course is intended as an introduction to this subject, which
occupies a central place in modern mathematics. We will learn
techniques for translating (certain) geometric problems into algebraic
ones. Once they are reformulated in algebraic language, one may unleash
the power of (commutative) algebra on them. Sometimes they even become
(at least in principle) amenable to treatment by a computer.
However, only fairly recently (since the 1970s) have algorithms (and
the computers powerful enough to run them!) become available to
actually carry out the necessary computations. The engine behind these
is Buchberger's algorithm
which is based on the notion of Gröbner
. (If you are curious about Gröbner bases already,
watch the movie!
The advent of these programs has enabled mathematicians to study
complicated examples which previously couldn't be investigated by hand,
in this way inspiring a wealth of new mathematics. It has also made the
subject interesting for computer scientists and
engineers, since many practical questions (e.g., in robotics) can be
stated as problems in algebraic geometry.
A good foundation in linear algebra (at the level of Math
) and the ability to formulate mathematical proofs. Some
knowledge of abstract
would be useful, but is not
strictly necessary. You should also be able to use
necessarily to program) a
computer. Please feel free to
contact me if you'd like to take this course, but are unsure whether
you have the right preparation.
course we will discuss systems of polynomial equations (ideals), their solution sets (varieties), and how these objects
can be effectively manipulated (algorithms).
We will try to cover
at least the first
four chapters of the book Ideals, Varieties, and Algorithms,
An Introduction to Computational Algebraic Geometry and Commutative
Algebra, Third Edition, by David
Little, and Donal
Springer, New York, 2007. The authors of the textbook
entertain a web
page with errata and software.
Other textbooks of interest (on reserve in the Science and Engineering Library):
- William W. Adams and Philippe Loustaunau, An Introduction to Gröbner Bases,
Graduate Studies in Mathematics 3,
American Mathematical Society, Providence, RI, 1994.
- Thomas Becker and Volker Weispfenning, Gröbner Bases: A Computational
Approach to Commutative Algebra, Graduate Texts in Mathematics 141, Springer, New York, 1993.
- David Cox, John Little, and Donal O'Shea, Using Algebraic Geometry, Graduate
Texts in Mathematics 185,
Springer, New York, 2005.
- David Eisenbud, Commutative
Algebra: with a View Toward Algebraic Geometry, Third Edition,
Graduate Texts in Mathematics 150,
Springer, New York, 1999.
- Gert-Martin Greuel and Gerhard Pfister, A Singular Introduction to Commutative
Algebra, Springer, New York 2002.
- Hal Schenck, Computational
Algebraic Geometry, London Mathematical Society Student Texts 85, Cambridge University Press,
- Bernd Sturmfels, Solving
Systems of Polynomial Equations, CBMS Regional Conference Series
in Mathematics 97, American
Providence, RI, 2002.
There will be a problem set assigned on a semi-regular basis,
handed out in class, and also posted on this website.
Homework Set 1
Homework Set 1, Solutions
Homework Set 2
Homework Set 2, Solutions
Homework Set 3
Homework Set 3, Solutions
Homework Set 4
Homework Set 4,
Homework Set 5
The problems will
range in difficulty from routine to more challenging. Completed
solutions are to be handed in at the beginning of class on
the due date specified on the respective homework set. No late homework will be accepted. However,
your lowest homework score will be dropped when computing your grade.
You are encouraged to work together on the exercises, but any graded
assignment should represent your own work.
Some of the homework problems (and the midterm exam) will involve the
use of computer algebra systems. No
previous experience with computer programming is assumed, but I expect
that you are able and willing to familiarize yourself with the use of
the program of your choice. For overall user-friendliness, I recommend
the general-purpose program Maple (which can do
algebra, calculus, graphics, and so on). If you prefer,
you may also use Macaulay
2, a software system written by Mike Stillman and Dan Grayson,
and explicitly designed to support computations in algebraic geometry
and commutative algebra. Both systems are available for most platforms
(Unix, Linux, Mac OS X, Window$, etc.). While Macaulay 2
is freely downloadable, Maple is not free. (Student licenses for
Version 11 run at $99.) However, Maple will be
accessible to students in the Program
in Computing Lab.
Other (free) algebraic geometry software
Information on how to use Maple for computations with
Gröbner bases may be found in Appendix C of the textbook. The
packages for Maple
discussed there (which may be downloaded from the website of the
authors) tend to be rather slow in comparison with a dedicated system
2. If you decide to use Macaulay 2,
you might want to consult a chapter
by Bernd Sturmfels
from a book on Macaulay 2,
illustrating its use of
for some of the computations in our textbook.
Exams and Paper
There will be a take-home
Midterm examination, due on Monday,
November 5, at the beginning of class. It will be handed out at
the previous class meeting.
Click here to download the Midterm
Students with conflicts with the
Midterm Exam in this course are responsible for discussing makeup
examinations with me no later than two weeks prior to the exam.
There will be no Final examination. However, students are
required to work on an independent project throughout the quarter. The
project will involve studying a class-related topic, and writing a
short summary paper on this subject, which will go through several
stages of revision. Your paper should be
self-contained and accessible to the other participants in the class.
Achieving this should take approximately 10 pages. At the end of the
course, you will read a referee report written by another student in
the class, and you will also write such a report about the paper of
another student. Here is a list of possible topics (in no particular
You may also suggest your own project topic.
- Gröbner bases over principal ideal domains.
- Gröbner bases for modules.
- Universal Gröbner bases.
- Gröbner bases in power series rings.
- Gröbner bases of ideals with finitely many zeroes.
- Modules, free resolutions, and the Hilbert Syzygy Theorem.
- Automatic Theorem Proving.
- Noncommutative Gröbner bases.
- Complexity of computing Gröbner bases.
- Gröbner fan and the state polytope.
- Generic initial ideals.
Here is more information about
You are strongly encouraged to type your assignments. In the Program in Computing Lab
you may access
at least two implementations
(also written as
), a mathematical text
processing system written by Donald Knuth.
The use of
TeX is simplified by LaTeX
), written by Leslie Lamport
If you wish to learn
LaTeX, there are many online
guides available, for example here
A good reference
book is Math into LaTeX by
If you know how to download and install new software on your computer,
you might also consider using the what-you-see-is-what-you-get text
editor TeXmacs written by Joris van der Hoeven.
It makes it unnecessary for you to learn the LaTeX typesetting
language while producing output of comparable quality. The program is
freely downloadable, available for various platforms,
able to import and export LaTeX
offers a plugin for Macaulay 2.
More detailed instructions about the project, including references for
the projects listed above, will be announced in the first week of the
Grading policy: Homework: 30%. Midterm Exam: 30%. Paper: 40%.
All scores and final grades will be available on the MyUCLA gradebook.
below for biographical information about some of the mathematicians
whose work we will encounter in this course:
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modified November 20, 2007.