Resolving Decompositions for Polynomial Modules

Abstract

Abstract:We introduce the novel concept of a resolving decomposition of a polynomial module as a combinatorial structure that allows for the effective construction of free resolutions. It provides a unifying framework for recent results of the authors for different types of bases.

Resolutions and Betti Numbers of Polynomial Modules via Involutive Bases

“…Here some progress has been made in recent years by Gerdt and Blinkov like the introduction of the alex division [16] and some generalisations of involutive bases [private communication]. In all cases it follows from the results in [8] that our approach remains valid, but we have not yet a working implementation.…”

“…In the recent paper [7], we developed a term order free version of this result using so-called marked bases which are of considerable interest for the construction of Hilbert schemes. An axiomatic framework unifying and generalising these different variants based on the novel concept of a resolving decomposition is contained in [8]. 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.…”

Complementary decompositions of monomial ideals and involutive bases

“…1. x 3 3 has maximal x 3 -degree in U. This implies immediately that no non-multiplicative power exists for it at the variable x 3 ; moreover, since its x 3 -degree is unique…”

Resolving Decompositions for Polynomial Modules

“…This case distinction comes from the differential (9). Summands appearing in it possess one of the following two possible forms:…”

“…This article represents a revised and expanded version of [9] which was presented at the conference Computer Algebra in Scientific Computing. Its main objective is to unify all our above-mentioned works in a general axiomatic framework.…”