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

We present algorithms to factorize homogeneous polynomials in the first q-Weyl algebra and the first Weyl algebra. By homogeneous we mean homogeneous with respect to a Z-grading on the first (q-)Weyl algebra which will be introduced in this article. It turns out that the factorization can be almost completely reduced to commutative univariate factorization with some additional easy combinatorial steps. We implemented the algorithms in the computer algebra system Singular (Decker et al. (2012), Greuel and Pfister (2007)), which provides algorithms and an interface to deal with noncommutative polynomials (Greuel et al. (2010)). The implementation beats currently available implementations dealing with factorization in the first Weyl algebra in speed and elegancy of the results. Furthermore it broadens the range of polynomials we can nowadays factorize in a feasible amount of time using a computer algebra system.