Modular Techniques for Intermediate Primary Decomposition

In Commutative Algebra and Algebraic Geometry, ''Primary decomposition'' is well-known as a fundamental and important tool. Although algorithms for primary decomposition have been studied by many researchers, the development of fast algorithms still remains a challenging problem. In this paper, we devise an algorithm for ''Strong Intermediate Primary Decomposition" via maximal independent sets by using modular techniques. In the algorithm, we utilize double ideal quotients to check whether a candidate from modular computations is an intersection of prime divisors or not. As an application, we can compute the set of associated prime divisors from the strong intermediate prime decomposition. In a naive computational experiment, we see the effectiveness of our methods.