On an installation of Buchberger's algorithm

The computation of Gröbner bases on a shared memory multiprocessor

“…• The useless pair elimination is done by using the Gebauer-Moller criterion which is defined in [16].…”

“…The first one, called Criterion 2 by Buchberger, [8], avoids certain reductions by forecasting that certain very particular S-polynomials can be reduced to 0. The second one is a recent refinement by Gebauer and Moller, [16], of Criterion 1 by Buchberger. They show that Ker(A) is in fact generated by a subset of the s(i,j) and, therefore, that only a subset of the S-polynomials S(i,j) need to be reduced.…”

“…Gebauer and Moller, [16], use the resolution of Taylor, [36], to describe their enhancement of the algorithm. Taylor studied the resolution of monomial ideals and found homogeneous bases for the different modules in this resolution.…”

“…The details on the way these considerations are actually used to update the set of pairs when a polynomial is added to the basis may be found in Gebauer and Moller, [16].…”

“…The algorithm it uses is described in a recent paper by Gebauer and Moller, [16], where a proof of correctness can also be found.…”