Janet Bases and Resolutions in CoCoALib

Resolving Decompositions for Polynomial Modules

“…Instead, our results should be considered as a "meta-machinery" which augments any concept of a basis that induces a resolving decomposition with an effective syzygy theory. As already mentioned above, we applied this "meta-machinery" already for the special case of Janet or Pommaret bases (including a concrete implementation in the COCOALIB) [1,6]. The case of marked bases is considered in great detail in [7,8] (the latter reference also describes a concrete implementation in COCOALIB for the case of ideals).…”

Resolutions and Betti Numbers of Polynomial Modules via Involutive Bases

“…However, in all these generalisations one looses Theorem 2.2, as one now only obtains simple bounds for the projective dimension and the regularity from the given basis. In the case of Janet bases, we showed in [2] with an explicit example that the difference in the breadth of the resolution can even become arbitrarily large. However, in benchmarks with ideals more typical for applications, it turned out that very often one still obtains an optimal "bounding box" and in the few cases where this was not the case, the difference was very small.…”

“…This library also contains an implementation of Janet bases by the authors. More detailed information about the implementation can be found in [2]. We performed a large number of benchmark computations using standard test examples in Gröbner bases theory.…”

“…We compared the performance of our implementation for both computing minimal resolutions and computing only Betti numbers with MACAULAY2 [14] and SINGULAR [15]. Detailed tables with timing and a lot of other data can be found in [1,2]. Here we only comment on some basic observations.…”

“…In this short note, we briefly survey some recent results of the authors on a novel approach to compute resolutions and Betti numbers; for more details and the proofs we refer to [1,2]. It is based on a combination of the theory of involutive bases and algebraic discrete Morse theory.…”