Groebner Bases - Winter Term 2024
This is a website for the courses Groebner Bases (368.007) and
Exercises for Groebner Bases (368.005). The exercises have
the official name "UE : Special Topics algebra and discrete mathematics".
Content
Groebner bases are a versatile tool that provides solutions
for many problems in commutative algebra.
They provide a common generalization
of Euclid's GCD-algorithm, Gaussian elimination and Hermite Decomposition
and allow, for instance, to do linear algebra over multivariate polynomial rings.
This lecture provides
an introduction to the basics of their theory.
Method of instruction
The course consists of a lecture (2hrs/week) + exercise classes
(1hr/week). For the exercise class, students will be asked
to solve homework problems and to present their solutions during the
exercise classes.
Exam
Lecture and Exercise classes are separate courses and can
be done independently. The suggestion is to take both.
-
Lecture: written test OR oral exam.
-
Exercise: participation + presentations.
Prerequisites
Basic knowledge in linear and abstract algebra, such as acquired
in the first year's courses in linear algebra of a Mathematics program,
or in the first year's mathematics courses in Physics, Computer Science, or AI.
Language of instruction
English.
Previous edition
GB course during winter term 2020.
Important Dates
If you wish to participate, but cannot take part because of the time of the lecture,
please contact me and come to the lecture on Oct 1 so that we can find a solution
that suits all.
- The lecture starts on October 1, 9:15, K 012 D.
- Exercise classes start on October 1, 11:00, K 012 D. On October 1st, there will
be a lecture instead of exercises.
- Written exams: First possibility on Feb 3, 2025, 9:00.
Lecture Notes
[pdf].
We will start with Chapter 9 (Resultants) on p.64.
Planned content in detail:
- Multivariate polynomial rings and algebraic geometry:
Resultants, ideals, solution sets of polynomial equations
(Nullstellensatz).
- Multivariate polynomial division:
Term orderings, monomial ideals, quotient and remainder
in a multivariate setting, the definition of Groebner bases.
- Construction of Groebner Bases:
S-polynomials and Buchberger's algorithm.
- Applications:
Elimination property, algebraic dependencies and dimension,
membership problems for ring and field extensions,
automated geometric theorem proving.
- Linear algebra in multivariate polynomial rings over Euclidean domains:
Gröbner bases of modules, strong Gröbner bases.
Exercises
Those students who also want to get credits for the additional exercise
classes (368.005) should work on the homework problems and discuss their
solutions during the exercise classes.
Warning: Some of the exercise problems may be challenging!
For the best grade, about 80% of the problems should be solved,
40% are required for a passing grade.
Literature
Basic references:
Additional material: