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So far we have explained Groebner basis computation in commutative polynomial rings. However Groebner basis can be considered in more general non-commutative rings. Weyl algebra is one of such rings and Risa/Asir implements fundamental operations in Weyl algebra and Groebner basis computation in Weyl algebra.
n dimensional Weyl algebra over a field
D=K<x1,…,xn,D1,…,Dn> is a non-commutative
algebra which has the following fundamental relations:
D is the ring of differential operators whose coefficients
are polynomials in
Di denotes the differentiation with respect to
According to the commutation relation,
D can be represented as a
In Risa/Asir, this type of monomial is represented
<<i1,…,in,j1,…,jn>> as in the case of commutative
That is, elements of
D are represented by distributed polynomials.
Addition and subtraction can be done by
but multiplication is done by calling
dp_weyl_mul() because of
the non-commutativity of
 A=<<1,2,2,1>>; (1)*<<1,2,2,1>>  B=<<2,1,1,2>>; (1)*<<2,1,1,2>>  A*B; (1)*<<3,3,3,3>>  dp_weyl_mul(A,B); (1)*<<3,3,3,3>>+(1)*<<3,2,3,2>>+(4)*<<2,3,2,3>>+(4)*<<2,2,2,2>> +(2)*<<1,3,1,3>>+(2)*<<1,2,1,2>>
The following functions are avilable for Groebner basis computation
in Weyl algebra:
Computation of the global b function is implemented as an application.
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