Algorithms for Primary Decomposition

Abstract

The primary decomposition methods of Eisenbud, Huneke and Vasconcelos are anal- ysed in detail providing proofs of important theorems and all the corresponding al- gorithms are programmed in the language of Singular. MOreover, we investigated the parallelization of two modular algorithms. In fact, we consider the modular com- putation of Gr ̈obner bases (resp. standard bases) and the modular computation of the associated primes of a zero–dimensional ideal and describe their parallel imple- mentation in Singular. The algorithms of Shimoyama and Yokoyama for primary decomposition of ideals are generalized to submodules of a free module over the polynomial ring in several variables with coefficients in a field. The algorithms are implemented in Singular.