Although computer algebra is a research field in its own, its main driving force comes from various fields of applications. These applications range from mathematics and computer science over physics and engineering up to applications to technical and industrial problems.
In this article I should like to show, by means of examples, the impact of computer algebra on two branches of pure mathematics: algebraic geometry and singularity theory. Today, algorithms, programmes and systems in computer algebra have reached a stage where it is possible to compute highly sophisticated mathematical objects such as moduli spaces and objects related to mixed Hodge structures (Sections 2 and 3). Moreover, computer algebra has been and is still successfully used in testing or disproving conjectures, or in computing interesting examples (Section 4). Finally, I shall report on recent experiments where different methods of computer algebra have been applied to symbolic-numerical solving of polynomial equations (Section 5), an important application of computer algebra to real life problems. Each of the sections contains a few unsolved problems, respectively projects, for further research.
The examples were chosen either from diploma theses of some of my students (T. Bayer, M. Schulze, M. Wenk), respectively from joint research projects together with C. Lossen and E. Shustin. All algorithms are implemented in the computer algebra system SINGULAR [GPS]. They are mainly based on Gröbner basis methods which were foundationally developed by Buchberger [Bu1], [Bu2] for polynomial rings. Subsequently, they have been extended to local and ``mixed'' rings in [GP2] for use in singularity theory.
Gröbner basis computations are, nowadays, implemented in all major general purpose computer algebra systems such as the big-M-systems ( Magma, Maple, Mathematica, MuPad) but also in special systems designed for use in commutative algebra and algebraic geometry ( CoCoA, Macaulay, SINGULAR). However, having just the possibility to compute Gröbner bases (w.r.t. a few monomial orderings) is, for applications to mathematical research problems, not much more than having the elementary numerical operations on a calculator for applications to engineering problems. Hence, I should like to emphasise the necessity to further develop packages and libraries to make the systems still more useful for the ``working mathematician''. Today, new and advanced algorithms can be built on already existing powerful procedures for computing, for example, free resolutions, Ext and Tor groups, sheaf cohomology, primary decomposition, ring normalisation, versal deformations, and many more (cf. [GPS]). The development of new algorithms provides, in addition, a better understanding and often even produces new theoretical insight, as has been the case, just to mention one example, for primary decomposition (cf. [GTZ], [EHV]). This has also been the case for some topics treated in the present article.
For more applications of computer algebra to algebraic geometry and singularity theory see [Gr2].
We assume the reader is familiar with the main notions of Gröbner bases (cf. [CLO1], [BW]).
I should like to thank T. Bayer, C. Lossen und M. Schulze for helping to prepare this article and H.M. Möller for useful comments concerning symbolic-numerical solving.
Development and implementation of algorithms on which this paper is based were supported by the DFG-Schwerpunkt ``Effiziente Algorithmen für diskrete Probleme und ihre Anwendungen`` and by the ``Stiftung Rheinland-Pfalz für Innovation'', which we kindly acknowledge.