Modular Algorithms for Computing Minimal Associated Primes and Radicals of Polynomial Ideals

Abstract

In this paper, we propose algorithms for computing minimal associated primes of ideals in polynomial rings over Q and computing radicals of ideals in polynomial rings over a field. They apply Chinese Remainder Theorem (CRT) to Laplagne's algorithm which computes minimal associated primes without producing redundant components and computes radicals. CRT reconstructs an object in a ring from its modular images in the quotient rings modulo some ideals. In Laplagne's algorithm, ideals are decomposed over rational function fields by regarding some variables as parameters. In our new algorithms, we compute the minimal associated primes and the radical of < φ(G) > for a given ideal I= < G >, where φ is a substitution map for a parameter. Then we construct candidates of the minimal associated primes and the radical of I by applying CRT for those of < φ(G) >'s. In order for this method to work correctly, the shape of each modular component must coincide with that of the corresponding component of the ideal for computations of minimal associated primes, and radicals of modular images of given ideals must coincide with modular images of radicals of given ideals for radical computations. The former is realized with a high probability because a multivariate irreducible polynomial over Q remains irreducible after a substitution of integers for variables with a high probability and the latter is realized except for a finite number of moduli.