Factoring linear partial differential operators in n variables

Journal of Symbolic Computation

2018

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It has been recently discovered by Bell, Heinle and Levandovskyy that a large class of algebras, including the ubiquitous G-algebras, are finite factorization domains (FFD for short).Utilizing this result, we contribute an algorithm to find all distinct factorizations of a given element f ∈ G, where G is any G-algebra, with minor assumptions on the underlying field.Moreover, the property of being an FFD, in combination with the factorization algorithm, enables us to propose an analogous description of the factorized Gröbner basis algorithm for G-algebras. This algorithm is useful for various applications, e.g. in analysis of solution spaces of systems of linear partial functional equations with polynomial coefficients, coming from G. Additionally, it is possible to include inequality constraints for ideals in the input.

“…This has been studied e.g. for partial q-differential, differential and difference operators in [12,13], where the Z n graded structure resp. a certain embedding has been utilized.…”

Section: Discussionmentioning

confidence: 99%

Section: Possible Improvementsmentioning

confidence: 99%

Section: Possible Improvementsmentioning

confidence: 99%

“…Then the Algorithm 1 can be modified along the lines of algorithms from [12,13], which we have also sketched above. Let us illustrate this approach by a concrete example.…”

Section: The Irreducible Elements Inmentioning

confidence: 99%

See 2 more Smart Citations

A factorization algorithm for G-algebras and its applications

Journal of Symbolic Computation

2018

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“…[CCH + 11], [HL13] and [GHL14]. Writing benchmarks in the field of computer algebra differs from other benchmarking tasks.…”

Section: Introductionmentioning

confidence: 99%

The SDEval benchmarking toolkit

ACM Commun. Comput. Algebra

2015

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In this paper we will present SDeval, a software project that contains tools for creating and running benchmarks with a focus on problems in computer algebra. It is built on top of the Symbolic Data project, able to translate problems in the database into executable code for various computer algebra systems. The included tools are designed to be very flexible to use and to extend, such that they can be easily deployed even in contexts of other communities. We also address particularities of benchmarking in the field of computer algebra.Furthermore, with SDEval, we provide a feasible and automatable way of reproducing benchmarks published in current research works, which appears to be a difficult task in general due to the customizability of the available programs.

Factorization of ℤ$$ \mathbb {Z}$$-Homogeneous Polynomials in the First q-Weyl Algebra

Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory

2017

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We present algorithms to factorize weighted homogeneous elements in the polynomial first Weyl algebra and q-Weyl algebra, which are both viewed as Z-graded rings. We show that factorization of homogeneous polynomials can be almost completely reduced to commutative univariate factorization over the same base field with some additional uncomplicated combinatorial steps. This allows to deduce the complexity of our algorithms in detail. Furthermore, we will show for homogeneous polynomials that irreducibility in the polynomial first Weyl algebra also implies irreducibility in the rational one, which is of interest for practical reasons. We report on our implementation in the computer algebra system Singular. For homogeneous polynomials, it outperforms currently available implementations for factoring in the first Weyl algebra -in speed as well as in elegancy of the results.