Math 480, Solving Polynomial Equations (3 credits), Spring 2008

• Office : Padelford C-438
• Phone : (206) 616 9374
• E-mail : rrthomas (at) u (dot) washington (dot) edu
• Office hours : M: 3:30-5, W: 2:30-4, F: 9:30-10:15 (in PDL C-438).

Course Information

• Description : This is an advanced undergraduate class on solving systems of polynomial equations. Many applications in science, engineering, statistics, applied math and social sciences involve models that use polynomials as opposed to differential equations or linear equations, and in many of these instances it is important to know how to solve the resulting systems and/or manipulate the equations involved. This class will address the mathematical and algorithmic aspects of three basic issues concerning polynomial systems:
  • finding complex roots
  • finding real roots
  • polynomial optimization over sets described by polynomial equations and inequalities.
  
  
  
• Lecture : SIEG Hall 225, MWF 10:30-11:20
  
  
• Textbook : There is no official textbook for this class. However, two excellent books on this topic which I will use quite a bit are:
  • Ideals, Varieties and Algorithms , by Cox, Little and O'Shea, Springer UTM, and
  • Using Algebraic Geometry, by Cox, Little and O'Shea, Springer GTM.
  Other (more advanced) references:
  • Solving Polynomial Equations, Sturmfels, CBMS 97, AMS.
  
  
  
• Software : There will be a big emphasis on computation in this class.
  
  Software Packages that will be used mainly:
  • SAGE
    William Stein's Math 480 this quarter is about doing mathematical computations using his software package SAGE. In the first two weeks of the quarter, he will teach the basics of using SAGE. You are encouraged to attend these lectures if you can.
  • Maple
  It would be useful for you to download all these packages onto your computers as soon as possible.
  
  
  
• Prerequisites : Officially, the only prerequisite for this class is a solid understanding of linear algebra and an interest in computations. However, as polynomial systems are inherently algebraic objects, we will quickly find ourselves wading neck deep in abstract algebra. We will also need various other tools from convex geometry, analysis, combinatorics and optimization. So, being open to all sorts of mathematics will be an advantage.
  
  
• Assignments : The grade for this class will be based on homework assignments and a portfolio.
  
  • Homework (60% of grade) There will be ten homework assignments, one each week. Each homework set will consist of a computational and theoretical part. The theoretical questions will typically involve writing proofs. I will expect proficiency in writing proofs.
  • Portfolio (40% of grade) Each student will be expected to create a portfolio which at the end of the quarter should consist of ten "perfect" theory problems and ten "perfect" computational problems from the homework sets. So this is roughly one theoretical and one computational problem from each of the ten homework sets. You may choose which problems to include but there should be at least four problems of each kind from the first and second half of the quarter. These portfolio problems will be critiqued by both your fellow students and me and you have the chance to rewrite them as many times as you like during the quarter. The portfolio is due in class on June 6.
  
  
  
• Grading
  • Problems All problems in this class are worth 2 points. You will get a "2" for a perfect solution, "1" for a mostly correct solution and "0" otherwise. This applies to all homework and portfolio problems.
  • Overall grade in the class Your overall grade in the class will be one of the following five possibilities: 4.0, 3.5, 3.0, 2.5 and 1.8. Think of this as A,B,C,D and F. You must score at least 60% of the total number of points to get a 2.5.

Demos

Homework Assignments