> Praktische Mathematik: Einführung in das symbolische Rechnen - SS 211

AG Algebra und Geometrie

# Content:

## Dates

• Lecture  ( KIS )

Weekday

Time

Room

Start

Tuesday

11.45 am - 01.15 pm

48-210

19.04.2011

Thursday

11.45 am - 01.15 pm

48-210

21.04.2011

• Exercise classes  ( KIS )

Weekday

Time

Room

Tutor

Start

Wednesday

11.45 am - 01.15 pm

11-207

26.04.2011

Wednesday

03.30 pm - 05.00 pm

48-438

26.04.2011

• Programming Session

Weekday

Time

Room

Tutor

Start

Wednesday

01.30 pm - 03.00 pm

48-419

26.04.2011

## Prerequisites

The lectures "Grundlagen der Mathematik" and "Algebraische Strukturen" are assumed.

## Overview

Most of mathematics is concerned at some level with setting up and solving equations, for example to model applications in science and engineering. In many cases this involves tedious computations which are difficult to get right or too extensive to be carried through by hand. Two mathematical disciplines, numerical analysis and, more recently, computer algebra originated from this problem.

Calculations in numerical analysis are carried through approximately. They are very efficient, but subject to rounding errors.

Calculations in computer algebra are carried through exactly. They are usually less efficient and not always applicable (it may not be possible to solve huge systems of equations in due time, the input data may only be given approximately, there are no exact ways of representing the solutions).

However, computer algebra methods often provide more mathematical insight. Through infinite precision arithmetic, they allow us to actually compute, for instance, in the ring of integers and in the field of rationals, in finite prime fields, in algebraic number fields, and in arbitrary Galois fields. In fact, there is a much larger variety of algebraic structures in which algebraic algorithms allow us to manipulate algebraic objects or the structures itself.

Computer algebra is interdisciplinary in nature, with links to quite a number of areas in mathematics, with applications in mathematics and other branches of science, and with constantly new and often surprising developments. Algebraic algorithms allow us in many cases of theoretical and practical interest to study explicit examples. In this way, pure mathematics becomes accessible to experiments.

Particular fruitful interactions unfold between computer algebra and algebraic geometry, number theory, and group theory. Algebraic algorithms open up new ways of accessing subareas of these key disciplines of mathematics, and they are fundamental to practical applications of the disciplines. Conversely, challenges arising in algebraic geometry, number theory, and group theory quite often lead to algorithmic breakthroughs.

This lecture gives an introduction into basic algorithms, with particular emphasis on algorithms for experiments in number theory, commutative algebra, and algebraic geometry, with applications in cryptography and robotics.

In addition to theoretical exercise sessions, there will be practical training sessions in programming and implementing some of the algorithms in the object-oriented programming language Python and in the computer algebra system SINGULAR, respectively.

## Exercise sheets for the exercise classes

At most three names are allowed to appear on the homework you hand in. Nevertheless you are encouraged to discuss in larger groups. There will be a new sheet of exercises every Thursday. You have to submit your solutions every subsequent Thursday, 11.45 am in the corresponding shelves in building 48, 2nd floor ( beside room 48-208 ).

Nr.

Remarks

1

Sheet 1

2

Sheet 2

Ex. 4 (b): N=100 is enough.

3

Sheet 3

4

Sheet 4

5

Sheet 5

6

Sheet 6

There was a typo in Ex. 3 (b): p are prime divisors of N-1.

7

Sheet 7

8

Sheet 8

Note that p>3 in Exercise 1(b).

9

Sheet 9

10

Sheet 10

11

Sheet 11

Note that there was a typo in the definition of the ideal in Ex 4.

## Exercise sheets for the programming session

These exercises are only for those, who attend the practical sessions!
Regularly, active and successful participation in the practical sessions is required to achieve a "Praktikumsschein".

Nr.

Remarks

1

Sheet 1

2

Sheet 2

3

Sheet 3

4

Sheet 4

5

Sheet 5

6

Sheet 6

7

Sheet 7

8

Sheet 8

9

Sheet 9

10

Sheet 10

## Credit points

Regularly, active and successful participation in the exercise classes is required to achieve an "Übungsschein". This includes scoring at least 40% of the exercise points and presenting own solutions on the black board.
You have to pass an oral exam at the end of the term in order to get credit points for this lecture.
If you are in doubt whether or not you need an Uebungsschein, please ask your academic advisor ("Fachstudienberater" in German). In case you study mathematics, you may find him here.

## Literature

• Cohen: A Course in Computational Algebraic Number Theory. Springer, 1993.
• Cox, Little, O'Shea: Ideals, Varieties and Algorithms, Springer, 2007.
• Forster: Algorithmische Zahlentheorie, Vieweg, 1996.
• Gerhard, von zur Gathen: Modern Computer Algebra. Cambridge Press, 2003.
• Greuel, Pfister: A SINGULAR introduction to Commutative Algebra. 2nd ed., Springer, 2007.
• Kaplan: Computer Algebra. Springer, 2008.
• Knuth: The Art of Computer Programming. Volumes 1,2,3, Addison-Wesley, 1998.
• Lidl, Niederreiter: Introduction to Finite Fields, Cambridge University Press, 1986.
• PYTHON documentation, 2011.
• SINGULAR 3-1-3 Online Manual, 2011.

## Slides and script

Slides illustrating the usage of Gröbner basis computations in geometry ( 2011-06-21 )

Some introductory lecture notes:

Introductory lecture notes ( 2011-05-02 )

Some slides giving examples of explicit computer algebra computations are available here.

Introductory slides from the first lecture ( 2011-04-22 )

Slides from the second lecture ( 2011-04-27 )

Wolfram Decker und Christian Eder

 2011-07-13 [top]