An algorithm for primary decomposition in polynomial rings over the integers

We present an algorithm to compute a primary decomposition of an ideal in a polynomial ring over the integers. For this purpose we use algorithms for primary decomposition in polynomial rings over the rationals resp. over finite fields, and the idea of Shimoyama-Yokoyama resp. Eisenbud-Hunecke-Vasconcelos to extract primary ideals from pseudo-primary ideals. A parallelized version of the algorithm is implemented in SINGULAR. Examples and timings are given at the end of the article.