Math 191: Algorithms for Elementary Algebraic Geometry

# General Information

Time and Place: MWF 1-1:50pm, Dodd Hall 162

Instructor: Matthias Aschenbrenner

Homepage: http://www.math.ucla.edu/~matthias

Office:
Mathematical Sciences Building 5614
Office Phone: (310) 206-8576
Office Hours: M 2:30-4pm, W 2:30-4pm, or by appointment.

# Description

Geometric objects can often be described as the solution sets of algebraic equations. Simple examples in three-dimensional space are curves like
 (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 equation x²-y²z²+z³=0).
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 equations 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 basis. (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.

# Prerequisities

A good foundation in linear algebra (at the level of Math 115A) and the ability to formulate mathematical proofs. Some knowledge of abstract algebra would be useful, but is not strictly necessary. You should also be able to use (though not 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 Text

 In this 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 Cox, John Little, and Donal O'Shea, 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, Cambridge, 2003.
• Bernd Sturmfels, Solving Systems of Polynomial Equations, CBMS Regional Conference Series in Mathematics 97, American Mathematical Society, Providence, RI, 2002.

# Homework

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, Solutions

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 includes CoCoA and Singular.

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 as Macaulay 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.

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 order):
1. Gröbner bases over principal ideal domains.
2. Gröbner bases for modules.
3. Universal Gröbner bases.
4. Gröbner bases in power series rings.
5. Gröbner bases of ideals with finitely many zeroes.
6. Modules, free resolutions, and the Hilbert Syzygy Theorem.
7. Automatic Theorem Proving.
8. Noncommutative Gröbner bases.
9. Resultants.
10. Complexity of computing Gröbner bases.
11. Gröbner fan and the state polytope.
12. Generic initial ideals.
You may also suggest your own project topic.

You are strongly encouraged to type your assignments. In the
Program in Computing Lab you may access at least two implementations of TeX (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 and here. A good reference book is Math into LaTeX by George Grätzer.

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 files, and 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 quarter.

Grading policy: Homework: 30%. Midterm Exam: 30%. Paper: 40%.

All scores and final grades will be available on the MyUCLA gradebook.

# Historical Information

Click below for biographical information about some of the mathematicians whose work we will encounter in this course:

Bruno Buchberger
L. E. Dickson
Wolfgang Gröbner
Grete Hermann
David Hilbert
Heisuke Hironaka