Factoring linear partial differential operators in n variables

In this paper, we present a new algorithm and an experimental implementation for factoring elements in the polynomial nth Weyl algebra, the polynomial nth shift algebra, and Zn-graded polynomials in the nth q_-Weyl algebra. The most unexpected result is that this noncommutative problem of factoring...

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Bibliographic Details
Published in:Journal of symbolic computation 2016-07, Vol.75, p.127-148
Main Authors: Giesbrecht, Mark, Heinle, Albert, Levandovskyy, Viktor
Format: Article
Language:English
Subjects:
Factorization
Linear partial differential operators
Non-commutative algebra
Singular
Weyl algebra
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Summary:In this paper, we present a new algorithm and an experimental implementation for factoring elements in the polynomial nth Weyl algebra, the polynomial nth shift algebra, and Zn-graded polynomials in the nth q_-Weyl algebra. The most unexpected result is that this noncommutative problem of factoring partial differential operators can be approached effectively by reducing it to the problem of solving systems of polynomial equations over a commutative ring. In the case where a given polynomial is Zn-graded, we can reduce the problem completely to factoring an element in a commutative multivariate polynomial ring. The implementation in Singular is effective on a broad range of polynomials and increases the ability of computer algebra systems to address this important problem. We compare the performance and output of our algorithm with other implementations in major computer algebra systems on nontrivial examples.
ISSN:0747-7171
1095-855X
DOI:10.1016/j.jsc.2015.11.011