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	NoncommutativePolynomials
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ExampleSkewPolynomial

Edit detail for ExampleSkewPolynomial revision 2 of 5

	1 2 3 4 5
Editor: hemmecke Time: 2015/02/18 23:39:44 GMT+0
Note:

changed:
-    Let's first consider differential
-    operator algebra
    Let's first consider a differential
    operator algebra.

added:

    Similarly, we can define a
    shift algebra.

added:

    The multivariate case is only
    sligthly more complicated.

    Let us here use a field as coefficient domain.

Computing with non-commutative polynomials

Univariate differential case

Let's first consider a differential operator algebra.

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Z ==> Integer

Type: Void

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P ==> UnivariatePolynomial('x, Z)

Type: Void

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sigma1: Automorphism P := 1

$\label{eq1}\mbox{\rm R - > R}$

(1)

Type: Automorphism(UnivariatePolynomial?(x,Integer))

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delta1: P -> P := D$P

$\label{eq2}\mbox{theMap (...)}$

(2)

Type: (UnivariatePolynomial?(x,Integer) -> UnivariatePolynomial?(x,Integer))

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S1 ==> UnivariateSkewPolynomial('X, P, sigma1, delta1)

Type: Void

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x1: S1 := 'x

$\label{eq3}x$

(3)

Type: UnivariateSkewPolynomial?(X,UnivariatePolynomial?(x,Integer),R -> R,theMap(DIFRING-;D;2S;1,303))

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X1: S1 := 'X

$\label{eq4}X$

(4)

Type: UnivariateSkewPolynomial?(X,UnivariatePolynomial?(x,Integer),R -> R,theMap(DIFRING-;D;2S;1,303))

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X1*x1

$\label{eq5}{x \ X}+ 1$

(5)

Type: UnivariateSkewPolynomial?(X,UnivariatePolynomial?(x,Integer),R -> R,theMap(DIFRING-;D;2S;1,303))

Univariate shift case

Similarly, we can define a shift algebra.

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xp: P := 'x

$\label{eq6}x$

(6)

Type: UnivariatePolynomial?(x,Integer)

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sigma2: Automorphism P := morphism((p: P): P +-> p(x+1), (p: P): P +-> p(x-1))

$\label{eq7}\mbox{\rm R - > R}$

(7)

Type: Automorphism(UnivariatePolynomial?(x,Integer))

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delta2: P -> P := (p: P): P +-> 0

$\label{eq8}\mbox{theMap (...)}$

(8)

Type: (UnivariatePolynomial?(x,Integer) -> UnivariatePolynomial?(x,Integer))

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S2 ==> UnivariateSkewPolynomial('X, P, sigma2, delta2)

Type: Void

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x2: S2 := 'x

$\label{eq9}x$

(9)

Type: UnivariateSkewPolynomial?(X,UnivariatePolynomial?(x,Integer),R -> R,theMap(*1;anonymousFunction;2;initial;internal))

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X2: S2 := 'X

$\label{eq10}X$

(10)

Type: UnivariateSkewPolynomial?(X,UnivariatePolynomial?(x,Integer),R -> R,theMap(*1;anonymousFunction;2;initial;internal))

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X2*x2

$\label{eq11}{\left(x + 1 \right)}\ X$

(11)

Type: UnivariateSkewPolynomial?(X,UnivariatePolynomial?(x,Integer),R -> R,theMap(*1;anonymousFunction;2;initial;internal))

Multivariate case

The multivariate case is only sligthly more complicated.

Let us here use a field as coefficient domain.

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$\label{eq12}1$

(12)

Type: PositiveInteger?