Computer Algebra

Core Course, 4+2

Basic Information

Lecturer:

Michael Sagraloff

Lectures:     Monday and Wednesday 12-14 (c.t.), Room 24 in E1 4 
Tutor:tba
Tutorials:will be fixed within the first week of the lecture
Prerequisites: 
  • Analysis I and Linear Algebra I or Mathematics for Computer Scientists I and II
  • Basic knowledge in Number Theory and Algebra is useful, but not required.
  • Programming experience in Maple (or another computer algebra system) is useful, but not required.

Description

We will provide an introduction into the most fundamental and ubiquitous algorithms in computer algebra. We further focus on topics related to geometric computing with (real) algebraic curves and surfaces.

  • numbers and arithmetics: school method vs. faster methods for multiplication, floating point arithmetic, interval arithmetic
  • polynomial arithmetics: division with remainder, fast multipoint evaluation, (asymptotically fast) Euclidean algorithm, greatest common divisor, factorization, comparison and representation of algebraic numbers.
  • polynomial root finding: Sturm sequences, Descartes algorithm, Newton-Raphson method, complex root finding.
  • modular arithmetic and modular algorithms: evaluation, interpolation, Chinese Remainder Algorithm, prime number tests.
  • discrete and Fast Fourier transformation: fast multiplication of polynomials, fast Taylor shift.
  • elimination theory and polynomial system solving: polynomials ideal, resultants and subresultant sequences, multivariate division with remainder, Gröbner bases, Cylindrical Algebraic Decomposition.
  • geometric algorithms: topology of algebraic curves and surfaces, arrangement computation.

Information and Rules

This is a theoretical core course for computer science students and an applied mathematics core course for mathematics students.

Successful completion (that is, solving the expected 50 % of the weekly exercises and passing the exam) is worth 9 ECTS credit points. Expect to be imposed a total workload of about 270 hours, distributed to 90 hours of attended lectures and 180 hours of private study.

This course is intended for graduate students and/or senior undergraduate students. It consists of two two-hour lectures and a two-hour tutorial session per week.

Grading:

The grade is determined by the final exam.

Exercises:

  • There will be an assignment sheet every week. We will discuss the exact rules for handing in exercises with the participants within the first week of the lecture – expect something similar to other theoretical computer science or math lectures.
  • We expect you to regularly attend and participate in the tutorial sessions.

Lectures

DateTopicReferenceHomework

Literature

  • Jürgen Gerhard and Joachim von zur Gathen: Modern Computer Algebra. Cambridge University Press, third edition, 2013, ISBN 9781107039032.
    Available in the Campus library of Computer Science and Mathematics.
  • Saugata Basu, Richard Pollack, Marie-Françoise Roy: Algorithms in Real Algebraic Geometry. Springer, 2003, ISBN 3-540-00973-6.
    Available for download here.
  • Yap, Chee: Fundamental Problems in Algorithmic Algebra. Oxford University Press, 2000, ISBN 0-195-12516-9.
    Preliminary version available for download here.
  • Richard P. Brent and Paul Zimmermann: Modern Computer Arithmetic. Cambridge University Press, 2010, ISBN 0-521-19469-5.
    Preliminary version available for download here.
  • Wolfram Koepf: Computeralgebra – Eine algorithmisch orientierte Einführung. Springer Verlag, 2006, ISBN 3-540-29894-0. In German.
    Available for download for students of the University of Saarland via SpringerLink.
  • Hal Schenck: Computational Algebraic Geometry. Cambridge University Press, 2003, ISBN 9780521829649.