Computer Algebra, Fall 2003.

Week 1. Note: Some of these documents are for Maple 5 and higher, some are only for Maple 6 and higher. I'll change that later today.

• We will look at this Maple worksheet on sequences in class.
• We will look at this worksheet on polynomials and the use of the quotes: ' ' in class. If there's enough time we'll also look at the following worksheets: worksheet on rational functions, worksheet on modular arithmetic. Study the remaining worksheets for next time.
• Homework week 1.
• Programming in Maple.

• Another introduction to Maple and some exercises to test yourself (in the form of a pdf-file) with some of the answers.
• Programming assignment: multiplication of polynomials (this contains the homework for week 2).
• A sorting algorithm (called merge sort) based on the "divide and conquer" approach.
• Multiplying polynomials with the "divide and conquer" approach: the Karatsuba algorithm (next week).
• The Euclidean Algorithm.

• On Tuesday we explain the Karatsuba algorithm and then practise programming in the classroom.
• Thursday: You can read the worksheet The Extended Euclidean Algorithm at home. To practise the method of "ansatz with undetermined coefficients" we will implement the computation of the polynomials s,t in a different way. Furthermore, we will work on "partial fraction decomposition" of rational functions. The homework for week 3 will be to do at least one out of these three: computing the s and t in the extended Euclidean Algorithm using the ansatz method, or partial fraction decomposition, or Karatsuba.

• Integration of rational functions. Homework week 4: do the exercise in this worksheet.
• More on integration of rational functions.

• Short introduction to resultants. For more on resultants see the course from fall 1999.
• The residue.
• More on residues.

• One more example on integration of rational functions.
• Differential fields, logarithms polynomial case (contains the homework week 6).

• Logarithms general case.
• Another example for a logarithmic extension.
• valuations, Liouvilles Principle.

• The exponential case example 1. Example 2
• Differential operators
• Exponents (contains an assignment (optional)).

• Solving polynomial equations, polynomial ideals: Groebner basis. Here is a text file that explains the notion of polynomial ideals.
• Buchberger's criterium (contains an assignment) and applications.
• in class assignments.