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

Albert Heinle () and Viktor Levandovskyy ()
Additional contact information
Albert Heinle: University of Waterloo, David R. Cheriton School of Computer Science
Viktor Levandovskyy: RWTH Aachen University, Lehrstuhl B für Mathematik

A chapter in Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory, 2017, pp 455-480 from Springer

Abstract: Abstract Factorization of elements of noncommutative rings is an important problem both in theory and applications. For the class of domains admitting nontrivial grading, we have recently proposed an approach, which utilizes the grading in order to factor general elements. This is heavily based on the factorization of graded elements. In this paper, we present algorithms to factorize weighted homogeneous (graded) elements in the polynomial first q-Weyl and Weyl algebras, which are both viewed as ℤ $${ \mathbb {Z}}$$ -graded rings. We show that graded polynomials have finite number of factorizations. Moreover, the factorization of such 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, which we prove to be polynomial-time. Furthermore, we show, that for a graded polynomial p, irreducibility of p in the polynomial first Weyl algebra implies its irreducibility in the localized (rational) Weyl algebra, which is not true for general polynomials. We report on our implementation in the computer algebra system Singular. For graded polynomials, it outperforms currently available implementations for factoring in the first Weyl algebra—in speed as well as in elegancy of the results.

Keywords: Factorization; Noncommutative factorization; Weyl algebra; q-Weyl algebra; 68W30; 16Z05; 68-04; 68W40 (search for similar items in EconPapers)
Date: 2017
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-70566-8_19

Ordering information: This item can be ordered from
http://www.springer.com/9783319705668

DOI: 10.1007/978-3-319-70566-8_19

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().