How to define a ring of differential operator

Hello,

i want to define and eventually construct a groebner basis for an Ideal in a Ring of differnetial operators,
for example K[x1,.......xn][D1,.........,Dn] which should somehow be possible with nc_algebra
(since the Ring is a g-algebra).

Since i can only give singular information about variables i would can define how the xi and Dj interact with each other,
but to compute a groebner basis, singular has to handle any element of K[x1,....,xn] as a coefficient .
But x1x2 has to have the same degree as for example x1.

I do not see a possibility how Singular will allow me to define the noncommutative properties of the ring and also
handle any element out of K[x1,...,xn] the way it needs to (for example giving me x1+x2 as leading coefficient of (x1+x2)D1).


Is there any possibility to solve my problem?

best regards,

kolja.

Use a block ordering (as ordering for the variables x_i, Dx_i)
which allows to consider the variables of the first Block as the main variables and the variables of the second block as "coeffcients".
See Gianni, P.; Trager, B.;Zacharias, G.: Gröbner bases and Primary Decomposition of Polynomial Ideals. Journal of Symbolic Computation. 1985. for details, here this trick is used to compute Groebner basesin K[x_i][y_i] but it works also in the non-commutative case.

Thank you very much, i will try that

the library ratgb.lib does this automatically and in a user-friendly way.

see e.g. http://www.singular.uni-kl.de/Manual/4-0-2/sing_813.htm