University of Illinois at Urbana-Champaign
College of Liberal Arts and Sciences
Department of Mathematics

Mathematics 321, section B
Symbolic Algebra
9-9:50 AM MWF
141 Altgeld Hall

Professor Daniel R. Grayson
Office: 357 Altgeld Hall
Phone: 3-6209

Introduction:

Modern computer algebra packages solve systems of polynomial equations using a basic algorithm that includes as special cases the technique for solving systems of linear equations through row reduction of matrices or elimination of variables and the algorithm due to Euclid for finding the greatest common divisor of two integers or of two polynomials in one variable. We study this algorithm and its applications to algebraic geometry and applied fields such as robotics.

There will be an opportunity to pursue projects in the computer lab with various symbolic algebra programs, although the main thrust of the course is to explain the algorithms, how they work, and what they do for you, in a classroom situation.

Chapter 6 of the book covers applications to robotics and automatic geometric theorem proving. For the application to robotics, one begins by expressing the constraints on the position of the components of a robot arm via a system of polynomial equations.

The text:

Ideals, Varieties, and Algorithms: An introduction to computational algebraic geometry and commutative algebra , by Cox, Little, and O'Shea. The term "algebraic geometry" refers to that part of modern mathematics which tries to understand the solutions to systems of polynomial equations by means of qualititative geometric methods. The term "commutative algebra" refers to the study of algebraic operations which satisfy the commutative law f*g = g*f. Multiplication of numbers and of polynomials satisfies this law, hence the relevance.

Prerequisites:

This course lists Math 317 as a prerequisite, but I am willing to waive that requirement in order to get this course off the ground. What I ask is that you have enough mathematical maturity not to be startled when we define a "ring" to be an abstract set with two operations, plus and times, satisfying certain axioms familiar to you from arithmetic, and we start considering subsets of rings, equivalence relations on rings, and so on. The first rings we encounter will consist of numbers or of polynomials, so they will not be exotic things.

Computer algebra systems:

I mention here the ones I know something about.

  • Here is the home page for Macaulay 2, which I have written with Michael Stillman. It is a computer program designed to support computations in algebraic geometry and computer algebra.
  • Here is some information on Mathematica, a popular graphical, numerical, and symbolical mathematics program which I helped write. The functions Solve[], Reduce[], and GroebnerBasis[] might be especially interesting for us.
  • Office hours:

    I will hold office hours at 12 on Monday, 11 on Wednesday, and 10 on Friday.

    The grading:

    Homework will be assigned and graded. You are expected to do the homework on your own. The homework will count for 20% of the grade.

    There will be two hour exams, each counting for 20% of the grade. The final exam will count for 40% of the grade, and will be cumulative, but will emphasize the material presented after the second hour exam.

    The hour exams will be held in class on February 25 and April 3.

    You may do an extra credit project to earn points worth up to 10% of the grade. Come see me to negotiate the topic and scope of the project. The project may involve extra reading and exposition, creative research, implementation of algorithms in software, or use of software to explore properties or applications of systems of equations.