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FriCAS can compute Groebner bases for noncommutative polynomial rings of solvable type (of category SolvableSkewPolynomialCategory?). Below we give example using partial differential operators:

fricas
(1) -> Pdo := PartialDifferentialOperator(Polynomial(Integer), Symbol)

\label{eq1}\hbox{\axiomType{PartialDifferentialOperator}\ } (\hbox{\axiomType{Polynomial}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{Symbol}\ })(1)
Type: Type
fricas
xx := D(x)$Pdo + y*D(z)$Pdo

\label{eq2}{y \ {D_{z}}}+{D_{x}}(2)
Type: PartialDifferentialOperator?(Polynomial(Integer),Symbol)
fricas
yy := D(y)$Pdo - x*D(z)$Pdo

\label{eq3}-{x \ {D_{z}}}+{D_{y}}(3)
Type: PartialDifferentialOperator?(Polynomial(Integer),Symbol)
fricas
L := xx*xx + yy*yy

\label{eq4}{{\left({{y}^{2}}+{{x}^{2}}\right)}\ {{D_{z}}^{2}}}+{{\left(-{2 \  x \ {D_{y}}}+{2 \  y \ {D_{x}}}\right)}\ {D_{z}}}+{{D_{y}}^{2}}+{{D_{x}}^{2}}(4)
Type: PartialDifferentialOperator?(Polynomial(Integer),Symbol)
fricas
gPak := NGroebnerPackage(Polynomial(Integer), IndexedExponents(Symbol), Symbol, Pdo)
The constructor NGroebnerPackage takes 3 arguments and you have given 4 .




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