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SandBoxDemoXFreeAlgebra

Edit detail for SandBoxDemoXFreeAlgebra revision 1 of 1

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Editor: pagani Time: 2020/01/30 21:19:46 GMT+0
Note:

changed:
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\begin{axiom}
-- Category: XFreeAlgebra(vl : OrderedSet, R : Ring) : Category 
-- ++ Author: Michel Petitot petitot@lifl.fr
-- ++ Date Created: 91
-- should be enough to impl TensorAlgebra(R-Module), R comm. ?
-- see also SandBoxFreeProduct by B.P.

B:=OrderedVariableList [b[i] for i in 1..5]

R:=Expression Integer
X:=XDistributedPolynomial(B,R)

b:=[a::X for a in enumerate()$B]

t0:=0$X
t1:=1$X
t2:=b.1*b.2
t3:=b.3*t2 - t2*b.3

vl:=varList t3

leadingMonomial t3
reductum t3
degree t3

t4:=x*t3+y^2*t2+z^3*t1
degree t4
leadingMonomial t4
degree t4

leadingCoefficient t4
coefficients t4

X has XFreeAlgebra(B,R)
-- ok
\end{axiom}

fricas (1) ->

Category: XFreeAlgebra(vl : OrderedSet, R : Ring) : Category -- ++ Author: Michel Petitot petitot@lifl.fr -- ++ Date Created: 91 -- should be enough to impl TensorAlgebra(R-Module), R comm. ? -- see also SandBoxFreeProduct by B.P.
B:=OrderedVariableList [b[i] for i in 1..5]

$\label{eq1}\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{b_{1}}, \:{b_{2}}, \:{b_{3}}, \:{b_{4}}, \:{b_{5}}\right]}\right)$

Type: Type

fricas

R:=Expression Integer

$\label{eq2}\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)$

Type: Type

fricas

X:=XDistributedPolynomial(B,R)

$\label{eq3}\hbox{\axiomType{XDistributedPolynomial}\ } \left({{\hbox{\axiomType{OrderedVariableList}\ } \left({\left[{b_{1}}, \:{b_{2}}, \:{b_{3}}, \:{b_{4}}, \:{b_{5}}\right]}\right)}, \:{\hbox{\axiomType{Expression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}}\right)$

Type: Type

fricas

b:=[a::X for a in enumerate()$B]

$\label{eq4}\left[{b_{1}}, \:{b_{2}}, \:{b_{3}}, \:{b_{4}}, \:{b_{5}}\right]$

Type: List(XDistributedPolynomial(OrderedVariableList([b[1],b[2],b[3],b[4],b[5]]),Expression(Integer)))

fricas

t0:=0$X

$\label{eq5}0$

Type: XDistributedPolynomial(OrderedVariableList([b[1],b[2],b[3],b[4],b[5]]),Expression(Integer))

fricas

t1:=1$X

$\label{eq6}1$

Type: XDistributedPolynomial(OrderedVariableList([b[1],b[2],b[3],b[4],b[5]]),Expression(Integer))

fricas

t2:=b.1*b.2

$\label{eq7}{b_{1}}\ {b_{2}}$

Type: XDistributedPolynomial(OrderedVariableList([b[1],b[2],b[3],b[4],b[5]]),Expression(Integer))

fricas

t3:=b.3*t2 - t2*b.3

$\label{eq8}{{b_{3}}\ {b_{1}}\ {b_{2}}}-{{b_{1}}\ {b_{2}}\ {b_{3}}}$

Type: XDistributedPolynomial(OrderedVariableList([b[1],b[2],b[3],b[4],b[5]]),Expression(Integer))

fricas

vl:=varList t3

$\label{eq9}\left[{b_{3}}, \:{b_{2}}, \:{b_{1}}\right]$

Type: List(OrderedVariableList([b[1],b[2],b[3],b[4],b[5]]))

fricas

leadingMonomial t3

$\label{eq10}-{{b_{1}}\ {b_{2}}\ {b_{3}}}$

Type: XDistributedPolynomial(OrderedVariableList([b[1],b[2],b[3],b[4],b[5]]),Expression(Integer))

fricas

reductum t3

$\label{eq11}{b_{3}}\ {b_{1}}\ {b_{2}}$

Type: XDistributedPolynomial(OrderedVariableList([b[1],b[2],b[3],b[4],b[5]]),Expression(Integer))

fricas

degree t3

$\label{eq12}3$

Type: PositiveInteger?

fricas

t4:=x*t3+y^2*t2+z^3*t1

$\label{eq13}{{z}^{3}}+{{{y}^{2}}\ {b_{1}}\ {b_{2}}}+{x \ {b_{3}}\ {b_{1}}\ {b_{2}}}-{x \ {b_{1}}\ {b_{2}}\ {b_{3}}}$

Type: XDistributedPolynomial(OrderedVariableList([b[1],b[2],b[3],b[4],b[5]]),Expression(Integer))

fricas

degree t4

$\label{eq14}3$

Type: PositiveInteger?

fricas

leadingMonomial t4

$\label{eq15}-{x \ {b_{1}}\ {b_{2}}\ {b_{3}}}$

Type: XDistributedPolynomial(OrderedVariableList([b[1],b[2],b[3],b[4],b[5]]),Expression(Integer))

fricas

degree t4

$\label{eq16}3$

Type: PositiveInteger?

fricas

leadingCoefficient t4

$\label{eq17}- x$

Type: Expression(Integer)

fricas

coefficients t4

$\label{eq18}\left[ - x , \: x , \:{{y}^{2}}, \:{{z}^{3}}\right]$

Type: List(Expression(Integer))

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X has XFreeAlgebra(B,R)

$\label{eq19} \mbox{\rm true}$

Type: Boolean