Algorithms for computing a primary ideal decomposition without producing intermediate redundant components

In Noro (2010) we proposed an algorithm for computing primary ideal decomposition by using the notion of separating ideal and showed that it can efficiently decompose several examples which are hard to decompose by existing algorithms. In particular the number of redundant components produced in the algorithm is zero or very small in many examples, but no theoretical explanation for the efficiency was made. In this paper we define a more sophisticated class of separating ideals: saturated separating ideal. By using this notion we modify the algorithm in Noro (2010) so that it directly outputs a minimal primary decomposition without producing any intermediate redundant component. By modifying the process of extraction of a primary component via pseudo-primary decomposition proposed in Shimoyama, Yokoyama (1996), we find a method for intermediate decomposition of an ideal and propose a variant of the new primary decomposition algorithm based on this intermediate decomposition. Our experiment shows that this variant efficiently decomposes many examples which are still hard to decompose even if we apply the original version of the new algorithm. Furthermore, in this algorithm we can bypass the computation of primary components and obtain directly the set of all associated primes of an ideal.