TU Kaiserslautern - Fachbereich Mathematik AG Algebra, Geometrie und Computeralgebra Dr. Janko Böhm

Tel.: +49 (0)631/205-2257 boehm@mathematik.uni-kl.de Zi. 431, Geb. 48 D-67663 Kaiserslautern

Computer Algebra

Summer Term 2017

News

  Lecture notes up to 20.07.2017.

  The final replacement lecture will take place on Thursday 20.07.17 from 8:20-9:50 in 48-438.

  Homework 13

  The second replacement lecture for the holidays will take place on Thursday 29.06.17 from 8:20-9:50 in 48-438.

  The frist replacement lecture for the holidays will take place on Thursday 22.06.17 from 8:20-9:50 in 48-438.

  Please fill out this Doodle to determine a date for the lectures replacing the ones on the holidays 25.05.17 and 15.06.17.

  Tutorial 2

  The exercise classes will begin on Wednesday 26.04.17 at 08:15 in 48-419.

  Please register online in the URM system.

  The lectures start on Thursday 20.04.17 at 13:45-15:15 in 48-438.

Goals

The lecture Computer Algebra will provide a systematic introduction to computational techniques in commutative algebra. This is the basis of modern computational algebraic geometry. Practical applications, some of which we will encounter in short examples during the lecture, include subjects like cryptography, robotics, bioinformatics, algebraic statistics, and even questions in theoretical physics like planetary motions and string theory.

The fundamental topics of the lecture are:
 Monomial orderings, division with remainder, normal forms, standard bases, Bucherger's and Mora's algorithm
 Ideal and module membership, radical membership, intersections, quotient, saturation, elimination
 Smith normal form
 Free resolutions, Hilbert's Syzygy Theorem
 Primary decomposition
 Hilbert series, Hilbert polynomial, Dimension theory, Noether normalization.

The lecture requires knowledge of the basic structures in algebra. Having attended the lecture "Commutative Algebra" is useful but not necessary. The topic of the lecture is strongly connected to the lecture "Algebraic Geometry", which you could attend before or afterwards.

Reference books

Work load

Module description for the MSc courses in Mathematics.

 4 SWS lectures 2 SWS exercises
 9 credit points

 60h contact hours for the lecture
 30h contact hours for the exercises
 180h self-study hours

Lectures

Lectures are Tuesdays and Thursdays, 13:45-15:15 in 48-438.

The classes begin on Thursday, 20.04.17.

The replacement lectures for the holidays will take place on Tuesday 22.06.17 from 8:20-9:50 in 48-436 and on Thurday 29.06.17 from 8:20-9:50 in 48-436.

Grading

There will be weekly problem sets distibuted every Thursday and due on the following Thursday. They will be collected at the beginning of the class in 48-438. Computer algebra sessions and program code should be handed as a printout and also via email to rottner@mathematik.uni-kl.de. The solutions will be graded and returned in the exercise class.

To obtain credit points you have to pass an oral exam at the end of the lecture period. In preparation to the exam you should also hand in your homework regularly. Likewise, the exercise class is not mandatory, but attending it regularly is strongly recommended.

If you only want to achieve an "Übungsschein" you have to attend the exercise classes and hand in your homework regularly. If you are in doubt whether you need an "Übungsschein", please ask your "Fachstudienberater" or the "Geschäftsführer" of the Mathematics Department.

We encourage you to team up to work on the exercises. It is also ok if teams of two students hand in a joint solution.

Exam

For the oral exams we will fix some dates to choose from in class. Then please send an email to boehm@mathematik.uni-kl.de to indicate whether the dates are ok for you and which of the dates you would prefer.

Please also register for the exam with Petra Bäsell (48-432).

Computer Algebra Systems

We will mainly use Singular, but will also have a look at other systems.

Singular

The open source computer algebra system Singular is being developed in Kaiserslautern. It is one of the leading systems for calculations in polynomial rings. Algorithms implemented in Singular deal with Gröbner bases, free resolutions, polynomial factorization, primary decomposition, and many other problems in commutative algebra.

You can download Singular here, read the online manual is here, and obtain the source code here.

Singular can be used without installation in a browser via the online interface.

Alternative systems are Macaulay2 and CoCoA.

GAP

We will have a quick look at the open source computer algebra system GAP, which is focussed on group theory dealing with calculation in the symmetric group:

GAP can be downloaded here and you can find the online help here.

On a Debian-like system you can install GAP by:
sudo apt-get install gap

Macaulay2

This is also an open source computer algebra with a similar functionality like Singular. It is being developed at the University of Illinois at Urbana-Champaign. Its logo shows a syzygy (of planets):

You can download Macaulay2 here, read the online manual is here, and obtain the source code here.

For the use in the Math-CIP-Pool you need a statically linked version. Do the following:

Download this file and that file.
In the directory where you stored the files do in the shell:

tar xzf Macaulay2-1.6-common.tar.gz
tar xzf Macaulay2-1.6-i686-Linux-Generic.tar.gz 

 To start Macaulay2 run:

 ./Macaulay2-1.6/bin/M2

Maple

In terms of a general purpose commercial computer algebra system, we will have a quick look at Maple. 

You can access Maple on the server lindb.rhrk.uni-kl.de of the Rechenzentrums with your usual user account.
From a Unix system log in by  ssh -X username@lindb.rhrk.uni-kl.de.

You can run the command line version by
module load maple/latest
maple
and the graphical user interface by
module load maple/latest
xmaple

To log in from a windows computer use, e.g., Putty. For the graphical user interface (and the plotting functionality of Singular) you need an X-Server, e.g., in Cygwin (with the packages xorg-server, xinit, xorg-docs, X-start-menu-icons, openssh). In the Cygwin-shell start the X-Server with startxwin.

Pari/Gp

Pari is an open source computer algebra system for number theory.

Here you can find the tutorial, a reference card, and the user guide.

Binary self installer for Windows.

Source code distribution for Linux and installation guide.

Pari/Gp can be used in an online interface.

Surfer

Surfer plots algebraic surfaces in 3-space. It is an easy to use front-end to the raytracing program Surf.

Exercises

Please register online in the URM system. Let us know if you have any issues with the date.

Tutorial 1 (for discussion in the exercise class on 26.04.17, does not have to be handed in):

Tutorial 2 (for discussion in the exercise class on 03.04.17, does not have to be handed in):

Assignment 1 (due on 04.05.17):

Assignment 2 (due on 11.05.17):

Assignment 3 (due on 18.05.17):

Assignment 4 (due until 26.05.17 in 48-427):

Assignment 5 (due on 01.06.17):

Assignment 6 (due on 08.06.17):

Assignment 7 (due until 16.06.17 in 48-427):

Assignment 8 (due on 22.06.17):

Assignment 9 (due on 29.06.17):

Assignment 10 (due on 06.07.17):

Assignment 11 (due on 13.07.17):

Assignment 12 (due on 20.07.17):

Assignment 13 (can be handed in):

Contact information

With respect to the lectures:

With respect to the exercise classes: