Meetings: MWF 11:00pm-11:50am, PHYSICS/GEOLOGY 140.

Instructor: Jesús A. De Loera.

email: deloera@math.ucdavis.edu

http://www.math.ucdavis.edu/~deloera/TEACHING/MAT165/165.html

Phone: (530)-754 70 29

Office hours: Monday and Wednesday 12:00pm-12:50pm or by appointment. My office is 3228 MSB. The TA for this class is Mr. Eddie Kim his office hours are Tues/Thurs from 10:30-11:30 Tu/Th (MSB 2131). We will be glad to help you with any questions, concerns, or problems.

Prerequisites and expectations: This class is intended for Math and CS majors in their junior or senior year. It is necessary that you have a solid idea of how to write proofs and true familiarity with computer programming (say as in ECS 30). In particular you will have to learn MAPLE and COCOA. If in doubt please ask me about it.

You are expected to work outside the classroom programming, thinking about the theorems and exercises, etc. I estimate a minimum of 3 hours work at home per lecture. The most important thing is what YOU learn by doing. Math and CS are not spectator sports!

Text: The text for this course is ``Ideals, Varieties, and Algorithms'' by D. Cox, J. Little and D. O'shea Textbook, 2nd edition, Springer 2003.

There will be also an optional text An introduction to MAPLE, by A. Heck, Springer, 2006. Although I personally think it is enough to read MAPLE's help and follow the class examples to learn enough of the language.

Description and Goals: This course has two goals:

1) To introduce undergraduate students to algorithmic mathematics. This is the part of mathematics dedicated to actually finding solutions explicitly (no existencial proof). We focus on mathematical algorithms where the answer is to be computed exactly. This is complementary to the area of numerical analysis (MATH 128ABC) where answers are computed with limited precision and error.

2) It is now undeniable that computers are useful tools for finding counterexamples, discover patterns, and even proof theorems! For example, the proof of the four color theorem, investigation of fractals, etc. Thus the second goal of the course is to learn how computers are useful tools for mathematical research, experimentation and can even help to generate formal proofs. Please read Computers and the meaning of Proof this article from Scientific American (a bit old now)

Course outline: We will cover a wide range of topics:

(week 1) Motivation, Introduction to MAPLE, The algebra of polynomials, the remainder theorem and roots. GCD of polynomials and the Euclidean Algorithm.

(week 2) The fundamental theorem of algebra. Real and rational roots, Sturm sequences and Descartes's rule of signs.

(week 3) Ideals and Varieties.

(week 4) Term orders and Multivariate Division Algorithm.

FIRST MIDTERM

(week 5-6) Groebner bases and Buchberger's algorithm

(week 7-8) Applications: Solutions of systems of polynomial equations, Ideal membership problem, Implicitization, applications to engineering.

(week 9-10) Polynomial systems and Computer Generated proofs, Radical ideals and Hilbert's nullstellensatz.

SECOND MIDTERM

Grading policy:

• There will be 2 midterms, first on November 6, 2006, the second on Monday December 4, each counting 35 points. I will also give 7 homework assignments (5 points each). For both midterms and homeworks I will drop the lowest score. Some homeworks may contain small programming-exploration problems. Finally there will be a final exam worth 40 points.

• As you see, there are 100 points possible in this course. I will assign grades based on the statistical information obtained (I compute the mean, standard deviation, etc.) and set letter grades according with those numbers. Nevertheless, it is my rule that at least 60 points are required to pass the class.

• Please remember there are NO make-up exams or late homeworks, instead I am dropping the lowest scores as a compensation of possible problems or emergencies. I will handle all grades via myucdavis grade system. This means that if you are a registered student at UC Davis you can access grade information for this class via the internet (check https://my.ucdavis.edu/ for details). This is a secure and private web page assigned to each student. You can see your standing in the class, important statistics on exams, and your final grade there! I will not disclose your grade in any other form.

  Class Computer Accounts

  To provide you with access to MAPLE please open an account at the mathematics department. Please go to:

  http://www.math.ucdavis.edu/comp/class-accts

  and follow the instructions. These accounts expire at the end of the quarter.

  Homeworks and Handouts

  homework 1 due October 11, 2006

  Here is the maple worksheet I showed in my introduction to the class

  Here is a piece of MAPLE code that should help you write the extended gcd computation. sample code

  homework 2 due October 23, 2006

  homework 3 due November 3, 2006

  review problems first midterm

  homework 4 due November 13, 2006

  homework 5 due November 20, 2006

  homework 6 due December 1, 2006

  Here is a piece of MAPLE code that should help you with using Grobner bases
  and solving systems of polynomials Grobner MAPLE code . You can also download a text version (not for xmaple) here .

  homework 7 due December 8, 2006

  review problems second midterm

  Here is the MAPLE code that solves the numerical problems, first in mw format,
  then in a text version (not for xmaple) text .

  Here are the MAPLE solutions for problems 3, 2B,2A of midterm 2. Here is the text version.

  FINAL's week information

  Our final will take place on the computer lab MSB, Saturday December 16 from 10:30am to 12:30pm. Please note the place of the exam and arrive to the main entrance of the building early (e.g. 10:20am) in case of problems with the door.

  Friday December 8 we will have a review session with the final in mind. Please bring all your questions! Here are some problems to motivate you bring your questions.

  During final's week I will have office hours every day, except Tuesday, from 8:30am to 10am and by appointment. The TA Eddie Kim will have office hours Tue Dec 12, 6-7 pm, Wed Dec 13, 12:45-1:45, Thr Dec 14, 12:30-1:30, and Fri Dec 15, 3:15-4:15 pm.