CoCoALib: A C++ Library for Computations in Commutative Algebra... and Beyond

Abbott, John; Bigatti, Anna Maria

doi:10.1007/978-3-642-15582-6_15

Cited by 76 publications

(155 citation statements)

References 4 publications

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“…We also implemented it in the COCOALIB [5]. To the best of our knowledge, this approach is the only one that is able to compute (even individual) Betti numbers without first determining a full resolution.…”

Section: Introductionmentioning

confidence: 99%

Resolving Decompositions for Polynomial Modules

Albert

Seiler

2016

Computer Algebra in Scientific Computing

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Section: Introductionmentioning

confidence: 99%

Resolving Decompositions for Polynomial Modules

Albert

Seiler

2016

Computer Algebra in Scientific Computing

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“…The computation has been done following the strategy outlined above, making use of Macaulay 2 [9] and CoCoA [1] for the computation of the Hilbert series of the Koszul homology H i (x, y, z, R(I 1 , I 2 )) and of the filtration allowing to decompose the Hilbert series as a sum of rational series with positive numerators and "partial" denominators similarly to what we have done in (3.1) and (3.2).…”

Section: Examplesmentioning

confidence: 99%

A remark on regularity of powers and products of ideals

Bruns

Conca

2017

Journal of Pure and Applied Algebra

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ABSTRACT. We give a simple proof for the fact that the Castelnuovo-Mumford regularity and related invariants of products of powers of ideals are asymptotically linear in the exponents, provided that each ideal is generated by elements of constant degree. We provide examples showing that the asymptotic linearity is false in general. On the other hand, the regularity is always given by the maximum of finitely many linear functions whose coefficients belong to the set of the degrees of generators of the ideals.

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“…Remarkably, the constructive nature of the main theorem proved in this paper (Theorem 1) may allow for the implementation of a symbolic package ( [15,16]) doing almost automatically all the lengthy and tedious calculations involved. Work in this direction is in progress.…”

Section: (Iv)mentioning

confidence: 99%

A Constructive Method for Standard Borel Fixed Submodules with Given Extremal Betti Numbers

Crupi

2017

Mathematics

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Let S be a polynomial ring in n variables over a field K of any characteristic. Given a strongly stable submodule M of a finitely generated graded free S-module F, we propose a method for constructing a standard Borel-fixed submodule M of F so that the extremal Betti numbers of M, values as well as positions, are preserved by passing from M to M. As a result, we obtain a numerical characterization of all possible extremal Betti numbers of any standard Borel-fixed submodule of a finitely generated graded free S-module F.

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CoCoALib: A C++ Library for Computations in Commutative Algebra... and Beyond

Cited by 76 publications

References 4 publications

Resolving Decompositions for Polynomial Modules

Resolving Decompositions for Polynomial Modules

A remark on regularity of powers and products of ideals

A Constructive Method for Standard Borel Fixed Submodules with Given Extremal Betti Numbers

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