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noncommutative Groebner bases

Edit detail for noncommutative Groebner bases revision 2 of 2

	1 2
Editor: hemmecke Time: 2024/01/09 17:15:47 GMT+0
Note: correct according to 96da3341640d9b207218400f485e66c453114241 (remove VarSet from NGroebnerPackage)

changed:
-gPak := NGroebnerPackage(Polynomial(Integer), IndexedExponents(Symbol), Symbol, Pdo)
gPak := NGroebnerPackage(Polynomial(Integer), IndexedExponents(Symbol), Pdo)

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)

$\label{eq1}\hbox{\axiomType{PartialDifferentialOperator}\ } \left({{\hbox{\axiomType{Polynomial}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \: \hbox{\axiomType{Symbol}\ }}\right)$

(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), Pdo)

$\label{eq5}\hbox{\axiomType{NGroebnerPackage}\ } \left({{\hbox{\axiomType{Polynomial}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \:{\hbox{\axiomType{IndexedExponents}\ } \left({\hbox{\axiomType{Symbol}\ }}\right)}, \:{\hbox{\axiomType{PartialDifferentialOperator}\ } \left({{\hbox{\axiomType{Polynomial}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}, \: \hbox{\axiomType{Symbol}\ }}\right)}}\right)$

(5)

Type: Type

fricas

groebner([L, xx])$gPak

$\label{eq6}\left[{{y \ {D_{z}}}+{D_{x}}}, \:{{D_{y}}^{2}}, \:{{y \ {D_{x}}\ {D_{y}}}-{D_{x}}}, \:{{D_{x}}^{2}}\right]$

(6)

Type: List(PartialDifferentialOperator?(Polynomial(Integer),Symbol))