PhD course in Gröbner bases

Abstractly, a Gröbner basis may be characterised as “a special case of a standard basis, i. e., a system of generators for a subobject of a filtered object or some similar object, such that the images of these objects generate the corresponding subobject of the graded associated object”. Concretely, it may be seen as “a sufficiently large set of generators to make problems of membership in ideals and other submodules tractible”. The way to find one in general centres around performing polynomial divisions with different divisors, and comparing the results.

Since finding a Gröbner basis makes many algebraic problems algiorithmically and efficiently tractible, they are a main tool in many modern computer algebra systems. On the other hand, finding Gröbner bases may have a prohibitive high complexity. Therefore, there has been a considerable interest in finding methods to improve the efficientcy in the calculation of Gröbner bases. We shall discuss some of the problems and suggested solutions.

Applications: Inter alia calculations of various algebraic objects and equation solving.

Recommended literature

An introduction to Gröbner Bases, Ralf Fröberg, John Wiley and Sons, 1997; ISBN 0 471 97442 0; however, this only partially will cover the content. The course will be held in English.

Prerequisites

A knowledge of algebra corresponding to the algebra part of the course “Commutative algebra and algebraic geometry” is indispensible. Having some acquaintance with non-commutative algebra and/or computer algebra programming might be advantageous. However, the course will formally start ‘from scratch’, summarily giving the theoretic background and definition of the topic. The more concrete presentation of the algorithms will come later. Finally, we shall treat some applications.