The termination of the F5 algorithm revisited

Abstract

The F5 algorithm [8] is generally believed as one of the fastest algorithms for computing Gröbner bases. However, its termination problem is still unclear. The crux lies in the non-determinacy of the F5 in selecting which from the critical pairs of the same degree. In this paper, we construct a generalized algorithm F5GEN which contain the F5 as its concrete implementation. Then we prove the correct termination of the F5GEN algorithm. That is to say, for any finite set of homogeneous polynomials, the F5 terminates correctly.

A survey on signature-based algorithms for computing Gröbner bases

“…4 Remark 7.22. Note that such attempts of generalizing the description of signature-based Gröbner basis algorithms have already been done, for example, in [58,68,69,76]. As we have already pointed out in the introduction of this section all of these characterizations are similar and included in our attempt using RB.…”

“…Extended F5 criteria uses different module monomial orders 8.3 [5] F5/2 adds field equations to the input systems for computations over 2 9.1 [36] bihomogeneous case uses maximal minors of Jacobian matrices to enlarge system of syzygies F5" [33] (2002) Extended F5 Criteria [5] (2010) F5GEN [68,69] (2013) F5/2 [36] (2003)…”

“…At least since Galkin's proof in [45] termination of F5 is clear. Several other publications include proofs of F5's termination, most of them are only slight variants or simplifications of Galkin's (see [68,69]), some are proving termination for slight variants of F5 (see [31]). The main idea is based on partitioning into sets…”

“…While the question of F5's termination was still an open one until recently [45,68,69], many new variants of F5 were introduced, for example, G2V [46] resp. GVW [47][48][49]83] or SB [70,71].…”

“…Moreover, first papers trying to classify all the different variants of signature-based Gröbner basis algorithms came up [30,31,58,68,69,76].…”