Most of mathematics is concerned at some level with setting up and solving various types of equations. Algebraic geometry is the mathematical discipline which handles solution sets of systems of polynomial equations. These are called algebraic sets.
Making use of a correspondence which relates algebraic sets to ideals in polynomial rings, problems concerning the geometry of algebraic sets can be translated into algebra. As a consequence, algebraic geometers have developed a multitude of often highly abstract techniques for the qualitative and quantitative study of algebraic sets, without, in fact, considering the equations at the first place. Modern computer algebra algorithms, on the other hand, allow us to manipulate the equations and, thus, to study explicit examples. In this way, algebraic geometry becomes accessible to experiments. The experimental method, which has proven to be highly successful in number theory, now also adds to the toolbox of the algebraic geometer.
In this lecture, we discuss some of the basic operations in geometry and describe their counterparts in algebra. We also indicate how the operations can be carried through using computer algebra methods, and give a number of explicit examples, worked out with the computer algebra system SINGULAR. In this way, the lecture connects nicely to courses on commutative algebra and computer algebra.
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Regular, 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.
Some chapters from the book by Decker and Schreyer
The book by Decker and Pfister
Slides from the lecture on May 24
Slides from the lecture on June 01
Slides from the lecture on June 18
Notes from the lecture on June 25
Slides from the lecture on June 28
Last change: July 23
Wolfram Decker
July 23, 2012 | [top] |