Practical Gröbner basis computation

Abstract

We report on our experiences exploring state of the art Gröbner basis computation. We investigate signature based algorithms in detail. We also introduce new practical data structures and computational techniques for use in both signature based Gröbner basis algorithms and more traditional variations of the classic Buchberger algorithm. Our conclusions are based on experiments using our new freely available open source standalone C++ library.

An analysis of inhomogeneous signature-based Gröbner basis computations

“…In [36] GVWHS is presented, a variant of GVW using the rewritable criterion of AP. This algorithm as well as the recently by Roune and Stillman in [32] presented SB algorithm coincide with our AP implementation.…”

“…In the last couple of years, several variants and optimizations in the class of signature-based algorithms, for example, F5C ( [17]), G2V ( [25]) or SB ( [32]) have been developed. Whereas the above mentioned publications focus mainly on the area of optimizing signature-based criteria for detecting useless critical pairs, a close look at the overall behaviour of signature-based computations in general is still missing.…”

On Gröbner Bases in the Context of Satisfiability-Modulo-Theories Solving over the Real Numbers

“…We base our implementation of data structures on [14] and use a compressed heap during the reduction and for storing S-pairs. However, the term and ideal representations are adapted based on the following observations: The number of variables in the system is usually high and, due to incrementality, we do not have a fixed bound on the number of variables at initialization.…”

“…. , n} | e i > 0} and i j < i j+1 for all 1 ≤ j < k. In our context, this representation seems more suitable than those from [14]. The degree is saved for fast access.…”

“…The basic features of this module are the simplification of equations and the check whether there are common zeros of the input equations. To tackle (2), we utilize some ideas from [12] and [14] to develop data structures that can handle a large number of variables and huge sets of sparse input polynomials, not necessarily of low degree, as they frequently occur in our setting. For (3), we further adapt Buchberger's algorithm in that we prune polynomials without real zeros in the Gröbner basis.…”