Factorization of Z-homogeneous polynomials in the First (q)-Weyl Algebra

TL;DR

This paper introduces algorithms for factorizing weighted homogeneous elements in the first polynomial Weyl and q-Weyl algebras, reducing the problem to univariate polynomial factorization and demonstrating practical efficiency.

Contribution

It provides the first algorithms for factorization of homogeneous polynomials in the first Weyl and q-Weyl algebras, with complexity analysis and an implementation in Singular.

Findings

Algorithms reduce factorization to univariate polynomial factorization.

Implementation in Singular outperforms existing methods in speed and results.

Irreducibility in the polynomial Weyl algebra implies irreducibility in the rational Weyl algebra.

Abstract

We present algorithms to factorize weighted homogeneous elements in the first polynomial Weyl algebra and $q$-Weyl algebra, which are both viewed as a $\mathbb{Z}$-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 \textsc{Singular}. It outperforms for homogeneous polynomials currently available implementations dealing with factorization in the first Weyl algebra both in speed and elegancy of the results.