An algorithm for primary decomposition in polynomial rings over the integers

TL;DR

This paper introduces a new algorithm for primary decomposition of ideals in polynomial rings over integers, leveraging existing algorithms over rationals and finite fields, with a parallel implementation in SINGULAR.

Contribution

It develops a novel algorithm combining existing methods for primary decomposition over different fields and implements a parallel version in SINGULAR.

Findings

Algorithm successfully computes primary decompositions for complex ideals.

Parallel implementation improves computational efficiency.

Examples demonstrate practical applicability and performance.

Abstract

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.