A criterion for detecting unnecessary reductions in the construction of Gröbner-bases

Computing representations for radicals of finitely generated differential ideals

“…The algorithm presented is much more efficient than the one of 1995. It applies for polynomial differential equations an analogue of the second criterion proved by Buchberger [1979] for Gröbner bases. Our implementation of this criterion was designed after the method of Gebauer and Möller [1988].…”

“…Buchberger [1979] established a few criteria which predict that some S-polynomials [Becker and Weispfenning, 1991, Definition 5.46] are reduced to zero without having to actually reduce them. They turn out to be very important in practice since most of the CPU time is spent in S-polynomials reductions.…”

The computation of Gröbner bases on a shared memory multiprocessor

“…We need a lemma, due to Zariski, on fields which are ring-finite over one of their subfields. A ring R is ring-finite over one of its subrings i2 7 , if there exist n elements of R, ri,... ,r n such that the ring homomorphism <f> from R' [x\,...,x n ] to R given by <f>(P) = P(^i,..., r n ) is onto (we will write that R = i2 ; [ri,... ,r n ]). We also say that R is module-finite over R f if R is a finitely generated iZ'-module.…”

“…Among the reasons of this meager speed-up, there are the criticisms that we formulate against Watt's approach, namely that the system does not keep track of the pairs already computed and that the reducers are tightly coupled. Moreover, Ponder does not seem to use the criteria described by Buchberger in [7] which avoid the computation of many useless S-polynomials. Ponder describes two other ways to parallelize Buchberger's algorithm.…”

Improved Computation of Involutive Bases

“…The notion of Gröbner bases was originally introduced in 1965 by Buchberger in his Ph.D. thesis and he also gave the basic algorithm to compute it [2,3]. Later on, he proposed two criteria for detecting superfluous reductions to improve his algorithm [1]. In 1983, Lazard [20] developed new approach by making connection between Gröbner bases and linear algebra.…”