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noncommutative Groebner bases

Edit detail for noncommutative Groebner bases revision 1 of 2

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Editor: test1 Time: 2015/06/30 18:55:01 GMT+0
Note:

changed:
-
FriCAS can compute Groebner bases
for noncommutative polynomial rings of solvable
type (of category SolvableSkewPolynomialCategory).
Below we give example using partial differential operators:
\begin{axiom}
Pdo := PartialDifferentialOperator(Polynomial(Integer), Symbol)
xx := D(x)$Pdo + y*D(z)$Pdo
yy := D(y)$Pdo - x*D(z)$Pdo
L := xx*xx + yy*yy
gPak := NGroebnerPackage(Polynomial(Integer), IndexedExponents(Symbol), Symbol, Pdo)
groebner([L, xx])$gPak
\end{axiom}

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  .