Computer Algebra

Core Course, 4+2


There will be no tutorial session tomorrow, January 4th.

Chapters 1-3 of the lecture notes are now online.

Today, December 18, the lecture will take place in Room 22 (MPII).

The typos in Exercise 1 in Exercise Sheet 4 is fixed now.

Exercise 4  in Exercise Sheet 4 is fixed now.

Chapter 1 of the lecture notes is now online.

We fixed a room for the tutorials. They will take place every Thursday (starting November 2nd) at 8.30-10.00 am in Room 24, Building E14

Since the Semester-kick-off Meeting will be held 4:30pm on 16 October, 2017,  our first lecture will start on 18 October, 2017.

Basic Information


Michael Sagraloff

Lectures:     Monday and Wednesday 12-14 (c.t.), Room 24 in E1 4 
Tutor:Anurag Pandey

Thursday 08:30 am - 10:00 am, Room 24 in E1 4 (first tutorial is on November 2nd)

  • 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.


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.


The grade is determined by the final exam. We will have an oral exam within the first two weeks of February.


  • There will be an assignment sheet every week, which will be posted on Monday. You have to hand in you solutions one week later before the lecture on Monday.
  • Solutions can be handed in as a group of at most 2 students
  • We expect you to regularly attend and participate in the tutorial sessions.


18.10.2017Introduction and Overview, School method and Karatsuba's method for integer multiplication 
23.10.2017Toom-Cook Multiplication 
25.10.2017Fixed-Point Arithmetic and Interval Arithmetic 
30.10.2017Box functions, Floating Point Arithmetic, Fast DivisionChapter 1: Basic Arithmetic
06.11.2017Fast Fourier Transform: Overview of the Schönhage-Strassen method for integer multiplication and Definition of Convolution, Primitive Roots of Unity, DFT 
08.11.2017DFT and FFT, Fast Convolution 
13.11.2017Schönhage Strassen method for the multiplication of integers and integer polynomials, Kronecker Substitution 
15.11.2017Fast Multiplication over arbitrary Rings, Fast Polynomial Division 
17.11.2017Fast Multipoint Evaluation and Applications 
20.11.2017Fast Multipoint Evaluation and Applications ctd., Fast Polynomial Arithmetic over C 
22.11.2017Fast Polynomial Arithmetic over C ctd., Integral Domains, Ideals

Chapter 2: The Fast Fourier Transform and Fast Polynomial Arithmetic   

27.11.2017 Noether Rings, Hilbert's Basis Theorem  
29.11.2017Factorial Rings, Gauss' Lemma 
04.12.2017Extended Euclidean Algorithm (Analysis and Definition), Square-Free Factorization 
06.12.2017Resultants (Definition and Properties) 
11.12.2017Mahler Measure, Root Separation Bounds 
18.12.2017Subresultants ctd., Structure Theorem for Subresultants

Chapter 3: The Extended Euclidean Algorithm and (Sub-) Resultants

20.12.2017Modular Computation, Prime Number Theorem, AKS Primality Test 


Exercise SheetDue Date
Exercise Sheet 1October 30
Exercise Sheet 2November 6
Exercise Sheet 3November 13
Exercise Sheet 4November 20
Exercise Sheet 5November 27
Exercise Sheet 6December 4
Exercise Sheet 7December 11
Exercise Sheet 8December 18
Exercise Sheet 9January 8
Exercise Sheet 10January 15
Exercise Sheet 11January 22


  • 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.