Additional Gröbner Basis Algorithms

Abstract

In §10 of Chapter 2 we discussed some criteria designed to identify situations where it is possible to see in advance that an S-polynomial remainder will be zero in Buchberger’s algorithm. Those unnecessary S-polynomial remainder calculations are in fact the main computational bottleneck for the basic form of the algorithm. Finding ways to avoid them, or alternatively to replace them with less expensive computations, is the key to improving the efficiency of Grobner basis calculation. The algorithms we discuss in this chapter apply several different approaches to achieve greater efficiency. Some of them use Grobner bases of homogeneous ideals or ideas inspired by the special properties of Grobner bases in that case. So we begin in §1 by showing that the computation of a homogeneous Grobner basis can be organized to proceed degree by degree. This gives the framework for Traverso’s Hilbert driven Buchberger algorithm, discussed in §2, which uses the Hilbert function of a homogeneous ideal to control the computation and bypass many unnecessary S-polynomial remainder calculations. We also show in §1 that the information generated by several S-polynomial remainder computations can be obtained simultaneously via row operations on a suitable matrix. This connection with linear algebra is the basis for Faugere’s F4 algorithm presented in §3. Finally, we introduce the main ideas behind signature-based Grobner basis algorithms, including Faugere’s F5 algorithm, in §4.