On the free resolution induced by a Pommaret basis

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

“…16 (2016) Sköldberg did not consider the question when a presentation of the special form (3) actually exists and how one could find it. Given an arbitrary presentation M ∼ = P m /U with a polynomial submodule U ⊆ P m , any resolution of U immediately yields one of M. We showed in [1] that any Pommaret basis of U automatically induces a presentation of U of the required form where the critical variables are just the non-multiplicative variables of the generators. Thus given a Pommaret basis of a submodule U, it have now two resolutions available: the one given by Theorem 2.1 and the one given by Sköldberg's approach with differential (4).…”

“…A closer analysis of it (detailed in [1]) reveals that one can derive very simple criteria which summands will lead to constant entries in the differential. This observation makes it feasable to compute directly only the constant part of the differential without any other terms in the differential.…”

“…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.…”