Factoring linear partial differential operators in n variables. (July 2016) - Giesbrecht, Mark; Heinle, Albert; Levandovskyy, Viktor
Abstract: In this paper, we present a new algorithm and an experimental implementation for factoring elements in the polynomial n th Weyl algebra, the polynomial n th shift algebra, and Z n -graded polynomials in the n th 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 Z n -graded, we can reduce the problem completely to factoring an element in a commutative multivariate polynomial ring. The implementation inSingular 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.