Computing Gröbner basis in the ring of differential operators

Example 10   Obtain the Gröbner basis of the ideal in the Weyl algebra

\begin{displaymath}{\bf Q } \langle x,y, \partial_{x} , \partial_{y} \rangle, \q...
...rtial}{\partial x},
\partial_{y} =\frac{\partial}{\partial y}
\end{displaymath}

generated by the differential operators

\begin{displaymath}x \partial_{x} + y \partial_{y} ,
\partial_{x} ^2 + \partial_{y} ^2
\end{displaymath}

in terms of the elimination order $ \partial_{x} , \partial_{y} > x,y $ by using the homogenized Weyl algebra.

%% gbdiff.sm1

[ (x,y) ring_of_differential_operators
  [[(Dx) 1 (Dy) 1]] weight_vector
 0
] define_ring

[ (x Dx + y Dy).
  (Dx^2 + Dy^2).
] /ff set

ff { [[(h). (1).]] replace homogenize} map /ff2 set

[ff2] groebner dehomogenize ::



Nobuki Takayama 2020-11-24