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.