I am looking for a software to perform calculations with modules over the algebra $R_n=\mathbb{C}(x_1\ldots x_n)\langle \partial_1\ldots\partial_n\rangle$ of differential operators with rational function coefficients. In the case of Weyl algebra $A_n$ with polynomial coefficients I know there is such functionality in Macaulay2 and Singular. Is there a system which allows working over $R_n$?

In principle I can factor my calculations through $A_n$, but my expressions are relatively simple when considered as polynomials in $R_n$ (i.e. I start with first-order diff. operators), but become very complicated when I reduce them to polynomial coefficients. I have tried going through Singular, but the computation is taking too much time. I expect that calculations directly over $R_n$ are actually much easier since the coefficients form a field. Is this expectation correct?

The sort of calculations I am looking for is computing syzygy modules and finding an expression for a given element of a module in terms of a given generating set. I guess it all boils down to computation of a Groebner basis.

Update: I found that OreModules package for Maple can in principle work with rational function coefficients. However, I can't find any accurate description of the algorithms used -- I have a feeling that it might be somehow re-using the polynomial coefficient code for rational computations. In any case, these packages fail to produce an answer on my machine in any reasonable time. I then become most interested in the question whether my expectation that the rational case is computationally simpler is correct.

Update 2: I have implemented the most naive version of Buchberger's algorithm over the rational function coefficients in Mathematica, and it appears to give a correct result in finite time, which suggests that either I am using OreModules package in a wrong way or that the package is indeed re-using the polynomial coefficient code.

  • $\begingroup$ I've found myself in the same situation. Would you be willing to share your code? $\endgroup$ – Jay Pantone Jun 2 '17 at 23:33
  • $\begingroup$ @JayPantone, I am not sure where I have this code. But as far as I remember, it was tailored very specifically to my problem and I am skeptical that anyone (me included) will be able to run it without pain and suffering. Unfortunately, I believe you will be better off implementing it yourself -- it is not that hard, the Buchberger's algorithm is pretty simple and works the same way over this algebra. $\endgroup$ – Peter Kravchuk Jun 3 '17 at 1:10

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.