Gröbner bases and primary decomposition of polynomial ideals

The computation of Gröbner bases on a shared memory multiprocessor

“…In particular, the list of applications that are mentioned is certainly not exhaustive. Some very important ones are not considered (e.g., the determination of the solutions of an algebraic equation and the recent article by Gianni, Trager and Zacharias, [29], on primary decomposition). There is much, therefore, that remains to be done.…”

“…In fact, the construction is still valid when I and J are ideals in the polynomial ring over certain rings (namely Noetherian commutative rings with identity in which linear equations are solvable) if the generators of J are not zero divisors. See [29] for details.…”

“…Gianni et al, [29], give a succinct description of this construction and of a more general one which allows us to compute a basis for the kernel of a given ring-homomorphism from P[t/i,... ,y p ] to P[xi,..., x n ]/I which is invariant on R . The paper by Shannon and Sweedler, [34], is more detailed and describes a subalgebra membership test as well.…”

“…The constructions described in this section are borrowed from Gianni, Trager and Zacharias, [29]. The constructions studying subalgebras of the polynomial ring are due to Shannon and Sweedler, [34].…”

Direct methods for primary decomposition

“…The increasing availability of symbolic algebra systems on computers and of efficient methods for 1)-3) has led to a renewed interest in the question of computing primary decompositions, as one sees from the work of Lazard (1982 and1985), Gianni et al (1988) (see also the references there), Bayer et al (1992), and Krick and Logar (1991). However these authors make use of the same basic strategy as Hermann, using PROJECTION to reduce to the one-polynomial case as before.…”