Winfried Neun

Very Large Gröbner Basis Calculations

The attempt to solve systems of polynomial equations with Grobner base techniques often leads to large problems which exceed the available computer resources with their requirements for cpu time or storage. The well-known reason for that is the swell of intermediate polynomials, which are generated during the basis calculation and which are in most cases not included in either the given set of polynomials or the resulting Grobner basis. In this paper two different approaches to overcome the problem are presented which benefit from the usage of parallel computers, namely the vectorization of the arbitrary precision integer arithmetic and the usage of decomposition techniques. Especially the decomposition approach, where applicable, leads to massive parallelism in the problem solution, which results in a breakthrough for several problems.