MathAction ExampleSkewPolynomial

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Note: 04-Apr-2021: This page is abandoned. It's new place is https://fricas.github.io/fricas-notebooks/FriCAS-SkewPolynomial.html

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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Q ==> Fraction UnivariatePolynomial('q, Z)

Type: Void

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VM ==> OrderedVariableList ['y, 'w, 'u]

Type: Void

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V  ==> OrderedVariableList ['Y, 'W, 'U]

Type: Void

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M ==> SparseMultivariatePolynomial(Q, VM)

Type: Void

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K := Fraction M

$\label{eq12}\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{SparseMultivariatePolynomial}\ } (\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{UnivariatePolynomial}\ } (q , \hbox{\axiomType{Integer}\ })) , \hbox{\axiomType{OrderedVariableList}\ } ([ y , w , u ])))$

(12)

Type: Type

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sigmay: Automorphism K := 1

$\label{eq13}\mbox{\rm R - > R}$

(13)

Type: Automorphism(Fraction(SparseMultivariatePolynomial?(Fraction(UnivariatePolynomial(q,Integer)),OrderedVariableList([y,w,u]))))

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yv: VM := 'y

$\label{eq14}y$

(14)

Type: OrderedVariableList([y,w,u])

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derivy(m: M): M == D(m, yv)

   Function declaration derivy : SparseMultivariatePolynomial(Fraction(
      UnivariatePolynomial(q,Integer)),OrderedVariableList([y,w,u]))
       -> SparseMultivariatePolynomial(Fraction(UnivariatePolynomial(q,
      Integer)),OrderedVariableList([y,w,u])) has been added to 
      workspace.

Type: Void

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deltay(k: K): K == D(k, derivy)

   Function declaration deltay : Fraction(SparseMultivariatePolynomial(
      Fraction(UnivariatePolynomial(q,Integer)),OrderedVariableList([y,
      w,u]))) -> Fraction(SparseMultivariatePolynomial(Fraction(
      UnivariatePolynomial(q,Integer)),OrderedVariableList([y,w,u]))) 
      has been added to workspace.

Type: Void

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wm: M := 'w

$\label{eq15}w$

(15)

Type: SparseMultivariatePolynomial?(Fraction(UnivariatePolynomial(q,Integer)),OrderedVariableList([y,w,u]))

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sigmaw1(k: K): K == eval(k, wm, wm+1)

   Function declaration sigmaw1 : Fraction(SparseMultivariatePolynomial
      (Fraction(UnivariatePolynomial(q,Integer)),OrderedVariableList([y
      ,w,u]))) -> Fraction(SparseMultivariatePolynomial(Fraction(
      UnivariatePolynomial(q,Integer)),OrderedVariableList([y,w,u]))) 
      has been added to workspace.

Type: Void

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sigmaw2(k: K): K == eval(k, wm, wm-1)

   Function declaration sigmaw2 : Fraction(SparseMultivariatePolynomial
      (Fraction(UnivariatePolynomial(q,Integer)),OrderedVariableList([y
      ,w,u]))) -> Fraction(SparseMultivariatePolynomial(Fraction(
      UnivariatePolynomial(q,Integer)),OrderedVariableList([y,w,u]))) 
      has been added to workspace.

Type: Void

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sigmaw: Automorphism K := morphism(sigmaw1, sigmaw2)

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Compiling function sigmaw1 with type Fraction(
      SparseMultivariatePolynomial(Fraction(UnivariatePolynomial(q,
      Integer)),OrderedVariableList([y,w,u]))) -> Fraction(
      SparseMultivariatePolynomial(Fraction(UnivariatePolynomial(q,
      Integer)),OrderedVariableList([y,w,u])))

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Compiling function sigmaw2 with type Fraction(
      SparseMultivariatePolynomial(Fraction(UnivariatePolynomial(q,
      Integer)),OrderedVariableList([y,w,u]))) -> Fraction(
      SparseMultivariatePolynomial(Fraction(UnivariatePolynomial(q,
      Integer)),OrderedVariableList([y,w,u])))

$\label{eq16}\mbox{\rm R - > R}$

(16)

Type: Automorphism(Fraction(SparseMultivariatePolynomial?(Fraction(UnivariatePolynomial(q,Integer)),OrderedVariableList([y,w,u]))))

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um: M := 'u; qQ: Q := 'q::Q; qQ1 := inv qQ

$\label{eq17}1 \over q$

(17)

Type: Fraction(UnivariatePolynomial(q,Integer))

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sigmau1(k: K): K == eval(k, um, qQ*um)$K

   Function declaration sigmau1 : Fraction(SparseMultivariatePolynomial
      (Fraction(UnivariatePolynomial(q,Integer)),OrderedVariableList([y
      ,w,u]))) -> Fraction(SparseMultivariatePolynomial(Fraction(
      UnivariatePolynomial(q,Integer)),OrderedVariableList([y,w,u]))) 
      has been added to workspace.

Type: Void

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sigmau2(k: K): K == eval(k, um, qQ1*um)$K

   Function declaration sigmau2 : Fraction(SparseMultivariatePolynomial
      (Fraction(UnivariatePolynomial(q,Integer)),OrderedVariableList([y
      ,w,u]))) -> Fraction(SparseMultivariatePolynomial(Fraction(
      UnivariatePolynomial(q,Integer)),OrderedVariableList([y,w,u]))) 
      has been added to workspace.

Type: Void

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sigmau: Automorphism K := morphism(sigmau1, sigmau2)$Automorphism(K)

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Compiling function sigmau1 with type Fraction(
      SparseMultivariatePolynomial(Fraction(UnivariatePolynomial(q,
      Integer)),OrderedVariableList([y,w,u]))) -> Fraction(
      SparseMultivariatePolynomial(Fraction(UnivariatePolynomial(q,
      Integer)),OrderedVariableList([y,w,u])))

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Compiling function sigmau2 with type Fraction(
      SparseMultivariatePolynomial(Fraction(UnivariatePolynomial(q,
      Integer)),OrderedVariableList([y,w,u]))) -> Fraction(
      SparseMultivariatePolynomial(Fraction(UnivariatePolynomial(q,
      Integer)),OrderedVariableList([y,w,u])))

$\label{eq18}\mbox{\rm R - > R}$

(18)

Type: Automorphism(Fraction(SparseMultivariatePolynomial?(Fraction(UnivariatePolynomial(q,Integer)),OrderedVariableList([y,w,u]))))

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deltawu(k: K): K == 0

   Function declaration deltawu : Fraction(SparseMultivariatePolynomial
      (Fraction(UnivariatePolynomial(q,Integer)),OrderedVariableList([y
      ,w,u]))) -> Fraction(SparseMultivariatePolynomial(Fraction(
      UnivariatePolynomial(q,Integer)),OrderedVariableList([y,w,u]))) 
      has been added to workspace.

Type: Void

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YV: V := index(1)$V; WV: V := index(2)$V;

Type: OrderedVariableList([Y,W,U])

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sigma(v: V): Automorphism K ==_
  if v=YV then sigmay else if v=WV then sigmaw else sigmau

   Function declaration sigma : OrderedVariableList([Y,W,U]) -> 
      Automorphism(Fraction(SparseMultivariatePolynomial(Fraction(
      UnivariatePolynomial(q,Integer)),OrderedVariableList([y,w,u])))) 
      has been added to workspace.

Type: Void

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delta(v: V): K -> K == if v=YV then deltay else deltawu

   Function declaration delta : OrderedVariableList([Y,W,U]) -> (
      Fraction(SparseMultivariatePolynomial(Fraction(
      UnivariatePolynomial(q,Integer)),OrderedVariableList([y,w,u])))
       -> Fraction(SparseMultivariatePolynomial(Fraction(
      UnivariatePolynomial(q,Integer)),OrderedVariableList([y,w,u])))) 
      has been added to workspace.

Type: Void

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T ==> SparseMultivariateSkewPolynomial(K, V, sigma, delta)

Type: Void

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y: T := 'y::K::T; w: T := 'w::K::T; u: T := 'u::K::T; q: T := qQ::K::T;

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Compiling function sigma with type OrderedVariableList([Y,W,U]) -> 
      Automorphism(Fraction(SparseMultivariatePolynomial(Fraction(
      UnivariatePolynomial(q,Integer)),OrderedVariableList([y,w,u]))))

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Compiling function delta with type OrderedVariableList([Y,W,U]) -> (
      Fraction(SparseMultivariatePolynomial(Fraction(
      UnivariatePolynomial(q,Integer)),OrderedVariableList([y,w,u])))
       -> Fraction(SparseMultivariatePolynomial(Fraction(
      UnivariatePolynomial(q,Integer)),OrderedVariableList([y,w,u]))))

Type: SparseMultivariateSkewPolynomial?(Fraction(SparseMultivariatePolynomial?(Fraction(UnivariatePolynomial(q,Integer)),OrderedVariableList([y,w,u]))),OrderedVariableList([Y,W,U]),theMap(*1;sigma;1;initial),theMap(*1;delta;1;initial))

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Y: T := monomial(1, index(1)$V, 1)

$\label{eq19}D_{Y}$

(19)

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W: T := monomial(1, index(2)$V, 1)

$\label{eq20}D_{W}$

(20)

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U: T := monomial(1, index(3)$V, 1)

$\label{eq21}D_{U}$

(21)

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t: T := y*w*u*q

$\label{eq22}q \ u \ w \ y$

(22)

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Y*t

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Compiling function deltay with type Fraction(
      SparseMultivariatePolynomial(Fraction(UnivariatePolynomial(q,
      Integer)),OrderedVariableList([y,w,u]))) -> Fraction(
      SparseMultivariatePolynomial(Fraction(UnivariatePolynomial(q,
      Integer)),OrderedVariableList([y,w,u])))

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Compiling function derivy with type SparseMultivariatePolynomial(
      Fraction(UnivariatePolynomial(q,Integer)),OrderedVariableList([y,
      w,u])) -> SparseMultivariatePolynomial(Fraction(
      UnivariatePolynomial(q,Integer)),OrderedVariableList([y,w,u]))

$\label{eq23}{q \ u \ w \ y \ {D_{Y}}}+{q \ u \ w}$

(23)

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W*t

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Compiling function deltawu with type Fraction(
      SparseMultivariatePolynomial(Fraction(UnivariatePolynomial(q,
      Integer)),OrderedVariableList([y,w,u]))) -> Fraction(
      SparseMultivariatePolynomial(Fraction(UnivariatePolynomial(q,
      Integer)),OrderedVariableList([y,w,u])))

$\label{eq24}{\left({q \ u \ w}+{q \ u}\right)}\ y \ {D_{W}}$

(24)

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U*t

$\label{eq25}{{q}^{2}}\ u \ w \ y \ {D_{U}}$

(25)

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)set message time on

(Y+y+1)^100;

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Time: 0.50 (EV) = 0.50 sec
(X1+x1+1)^100;

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

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Time: 0.25 (EV) = 0.25 sec
(X2+x2+1)^100;

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

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Time: 0.82 (IN) + 0.70 (EV) + 0.11 (OT) = 1.63 sec

Subject: Be Bold !!

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