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Let $K$ be a field, $\partial_i$ be commuting derivations on $K$, and consider the ring $R=K[\partial_1\ldots \partial_n]$ (it is implicitly assumed that the derivations do not commute with elements of $K$ and the commutation relation is the obvious $\partial_i k - k\partial_i=\partial_i(k)$).

For example, $R$ can be the ring of differential operators with rational function coefficients, etc.

It appears that much of the theory of Groebner bases for polynomial rings $S=K[x_1\ldots x_n]$ can be trivially restated for $R=K[\partial_1\ldots \partial_n]$, since when we work with a monomial order, we usually track only what happens to the leading monomial, and it is the same for $R$ and $S$. I am pretty sure that this is also true for modules over $R$. (If I am wrong and this is not true, an explanation of this will be a perfect answer to this question:) )

Is there a reference which restates the standard results over $S$ for modules over $R$?

Most of the references I have found deal with the more(?) complicated case of polynomial coefficients. Other than that, I am aware of two references, both by Nobuki Takayama, "Gröbner basis and the problem of contiguous relations" and a chapter in "Gröbner Bases: Statistics and Software Systems". The first basically introduces a version of Buchberger's algorithm for modules over $R$ and the second briefly goes over the very basic facts for ideals in $R$.

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  • $\begingroup$ Off the top of my head Saito, Sturmfels, Takayama, Grobner Deformations of Hypergeometric Differential Equations comes to mind. Have you had a look at it, althoug it also deals wth the more general case? $\endgroup$ – Michael Bächtold May 6 '16 at 6:40
  • $\begingroup$ @MichaelBächtold I have had a look at it, but I was under the impression that they work only with Weyl algebra. I am not sure I understand what you mean by the more general case. To provide some motivation, I am interested in the case of field coefficients because from physicist point of view this is a) often not even a restriction b) is more general since instead of rational functions I might need to use functions from some algebraic class and c) since only the derivatives contribute to the "degree of the polynomial", it appears to be computationally simpler in some situations. $\endgroup$ – Peter Kravchuk May 6 '16 at 19:31
  • $\begingroup$ By the more general case I though you were referring to polynomial coefficients. Are the results for that general case not applicable also when the polynomial ring has zero variables, hence for fields? $\endgroup$ – Michael Bächtold May 7 '16 at 6:15
  • $\begingroup$ @MichaelBächtold, that would be differential operators with constant coefficients, or am I missing something? $\endgroup$ – Peter Kravchuk May 8 '16 at 4:38
  • $\begingroup$ The base field of the polynomial ring (what you call constants) could in principle also be a field of rational fouctions. But I don't know which algorithmic properties Saito et.al. demand of the base field. $\endgroup$ – Michael Bächtold May 8 '16 at 5:50
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Elaborating on Michael's comments and your concerns about Saito, Sturmfels, Takayama, "Grobner Deformations of Hypergeometric Differential Equations", the case this book addresses is the case of the Weyl algebra over a characteristic 0 field $k$. I.e., the noncommutative polynomial ring $$D=k[x_1,\ldots, x_n, \partial_1,\ldots, \partial_n]$$, subject to the commutator relations $[\partial_i, x_j]=\delta_{ij}$. There's also a few points at which they address the larger ring $$k(x_1,\ldots, x_n)\otimes_{k[x_1,\ldots, x_n]} D.$$ Is this what you're looking for?

As for online things, some of the above is also covered in Oaku and Takayama's "Algorithms for D-modules —restriction, tensor product, localization, and local cohomology groups".

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  • $\begingroup$ Yes, I'm looking precisely for that second ring in your answer. Thank you the second reference, I will have a look at it. $\endgroup$ – Peter Kravchuk May 10 '16 at 3:36

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