Computing Gröbner bases on the Weyl algebras over fields with valuations

The computational aspect of tropical Gröbner bases for a polynomial ring K[x] with respect to tropical term orders studied by Chan and Maclagan in 2019 is extended to the Weyl algebra Dn(K), where K is a field with a valuation. The term order in this paper is not only an extension of the tropical term order on K[x] by Chan and Maclagan, but also of the tropical term order on K[x] studied by Vaccon et al. (2021). Due to the involvement of the valuations of term coefficients, this term order is not well-ordering. Therefore, a suitable division algorithm with respect to this term order is needed. This algorithm holds only for homogeneous operators, so utilizing the homogenized Weyl algebra is required. A computation example and an implementation in Risa/Asir Computer Algebra System are also presented in this paper.