Constructive D-Module Theory with Singular

Andres, Daniel; Brickenstein, Michael; Levandovskyy, Viktor; Martín-Morales, Jorge; Schönemann, Hans

doi:10.1007/s11786-010-0058-x

Cited by 15 publications

(19 citation statements)

References 34 publications

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“…A set of generators of the annihilator ideal Ann A 5 [s] (G s ) can be derived in SINGULAR [24] with algorithms introduced in [4]. Using the command SannfsBM we obtain a set of 13 generators.…”

Section: C1 From Annihilators To Integral Relationsmentioning

confidence: 99%

Feynman integral relations from parametric annihilators

et al. 2018

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We study shift relations between Feynman integrals via the Mellin transform through parametric annihilation operators. These contain the momentum space integration by parts relations, which are well-known in the physics literature. Applying a result of Loeser and Sabbah, we conclude that the number of master integrals is computed by the Euler characteristic of the Lee-Pomeransky polynomial. We illustrate techniques to compute this Euler characteristic in various examples and compare it with numbers of master integrals obtained in previous works. for interesting discussions and helpful comments on integration by parts for Feynman integrals, Jørgen Rennemo for suggesting literature relevant for section 3.1 and Viktor Levandovskyy for communication and explanations around D-modules and in particular their implementation in SINGULAR.Thomas Bitoun thanks Claude Sabbah for feedback and bringing the work [58-60] to our attention. Thomas acknowledges funding through EPSRC grant EP/L005190/1. Christian Bogner thanks Deutsche Forschungsgemeinschaft for support under the project BO 4500/1-1.This research was supported by the Munich Institute for Astro-and Particle Physics (MIAPP) of the DFG cluster of excellence "Origin and Structure of the Universe". Furthermore we are grateful for support from the Kolleg Mathematik Physik Berlin (KMPB) and hospitality at Humboldt-Universität Berlin.Images in this paper were created with JaxoDraw [11] (based on Axodraw [93]) and FeynArts [36].

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Section: C1 From Annihilators To Integral Relationsmentioning

confidence: 99%

Feynman integral relations from parametric annihilators

et al. 2018

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“…, x n is an element of x , that is, a word in the finite alphabet x. A standard monomial is a monomial of type x α = x α 1 1 · · · x αn n , where α = (α 1 , . .…”

Section: Preliminariesmentioning

confidence: 99%

“…More generally, given polynomials f 1 , . [1]). For our tests here, we use the method of Briançon and Maisonobe which can be described as follows: Consider the rth shift algebra S r (K) = s 1 .…”

Section: Examples From D-module Theory Involving the Weyl Algebramentioning

confidence: 99%

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Modular Techniques for Noncommutative Gröbner Bases

et al. 2019

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In this note, we extend modular techniques for computing Gröbner bases from the commutative setting to the vast class of noncommutative G-algebras. As in the commutative case, an effective verification test is only known to us in the graded case. In the general case, our algorithm is probabilistic in the sense that the resulting Gröbner basis can only be expected to generate the given ideal, with high probability.We have implemented our algorithm in the computer algebra system Singular and give timings to compare its performance with that of other instances of Buchberger's algorithm, testing examples from D-module theory as well as classical benchmark examples. A particular feature of the modular algorithm is that it allows parallel runs.

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“…Singular:Plural has implementations of many algorithms for D-module theory as well [4], including multivariate Weyl closure of a left ideal, polynomial/rational solutions and various local invariants for singularities. 6.3.4.…”

Section: Appendix On Computer Algebra Systems (By V Levandovskyy)mentioning

confidence: 99%