Course Description
Computer algebra is a very active area of research in the intersection of pure mathematics and computer science. The goal of the course is to give an introduction to the basic methods of computer algebra. All materials will be built from the basics and presented in a down-to-earth fashion. Important applications of computer algebra include cryptography, error correcting codes, image processing, kinematics, computer aided design and graphics.

I will try to cover the following topics:

• Algorithms, "Big Oh" notation, complexity classes
  
• Fundamental algorithms on integers and polynomials:
  representation, addition, multiplication and division with remainder
• Computation of the greatest common divisor of integers and polynomials:
  • Euclidean algorithm
  • Resultants and subresultants
  • Applications to error correcting codes
• Solution of multivariate polynomial systems
  • Polynomial ideals, membership problem
  • Groebner bases
  • Buchberger's algorithm
  • Applications
• Factoring polynomials:
  • Over finite fields
  • Application to finding roots
  • Hensel lifting
  • Short vectors in lattices
  • Application to breaking cryptosystems
    

We will use the following textbook:
Joachim von zur Gathen and Jurgen Gerhard: Modern Computer Algebra