-- -*- M2-comint -*- hash: 2931239423694942168

i1 : A = QQ<|x,y,z|>

o1 = A

o1 : FreeAlgebra

i2 : f = y*z + z*y - x^2

        2
o2 = - x  + y*z + z*y

o2 : A

i3 : g = x*z + z*x - y^2

            2
o3 = x*z - y  + z*x

o3 : A

i4 : h = z^2 - x*y - y*x

                    2
o4 = - x*y - y*x + z

o4 : A

i5 : B = A/ideal{f,g,h}
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.

o5 = B

o5 : FreeAlgebraQuotient

i6 : centralElements(B,3)

o6 = | -y3+yzx-zyx+z3 |

             1      1
o6 : Matrix B  <-- B

i7 : j = z^3+y*z*x-z*y*x-y^3

        3                    3
o7 = - y  + y*z*x - z*y*x + z

o7 : B

i8 : isCentral j

o8 = true

i9 : apply(5,i->isLeftRegular(j,i+1))
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.

o9 = {true, true, true, true, true}

o9 : List

i10 : normalElements(B,3,n)

o10 = ({}, {| n_(z^2*y) n_(z^2*x) n_(z*y*z) n_(z*y^2) n_(z*y*x)+n_(z^3)
      -----------------------------------------------------------------------
      n_(y*z*y) n_(y*z*x)-n_(z^3) n_(y^3)+n_(z^3) n_(y^2*x) |})

o10 : Sequence

i11 : ncBasis(3,B)

o11 = | y2x y3 yzx yzy zyx zy2 zyz z2x z2y z3 |

              1      10
o11 : Matrix B  <-- B

i12 : C = skewPolynomialRing(QQ,(-1)_QQ,{x,y,z,w})

o12 = C

o12 : FreeAlgebraQuotient

i13 : R = QQ[q]/ideal{q^4+q^3+q^2+q+1}

o13 = R

o13 : QuotientRing

i14 : B = skewPolynomialRing(R,q,{x,y,z,w})

o14 = B

o14 : FreeAlgebraQuotient

i15 : x*y == q*y*x

o15 = true

i16 : use C

o16 = C

o16 : FreeAlgebraQuotient

i17 : sigma = map(C,C,{y,z,w,x})

o17 = map (C, C, {y, z, w, x})

o17 : RingMap C <-- C

i18 : D = oreExtension(C,sigma,a)
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.

o18 = D

o18 : FreeAlgebraQuotient

i19 : isNormal a

o19 = true

i20 : apply(5,i-> isLeftRegular(a,i+1))
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.

o20 = {true, true, true, true, true}

o20 : List

i21 : sigmaD = normalAutomorphism a

o21 = map (D, D, {y, z, w, x, a})

o21 : RingMap D <-- D

i22 : normalElements(sigmaD,1)

o22 = | a |

              1      1
o22 : Matrix D  <-- D

i23 : normalElements(sigmaD,2)

o23 = 0

               1
o23 : Matrix QQ  <-- 0

i24 : normalElements(D,2,P)

         2
o24 = ({a }, {| P_(a*w) P_(a*z) P_(a*y) P_(a*x) P_(w*z) P_(w*y) P_(w*x)
      -----------------------------------------------------------------------
      P_(z*y) P_(z*x) P_(y*x) P_(z^2)^2-P_(y^2)P_(w^2)
      -----------------------------------------------------------------------
      P_(y^2)P_(z^2)-P_(x^2)P_(w^2) P_(x^2)P_(z^2)-P_(w^2)^2
      -----------------------------------------------------------------------
      P_(y^2)^2-P_(w^2)^2 P_(x^2)P_(y^2)-P_(z^2)P_(w^2)
      -----------------------------------------------------------------------
      P_(x^2)^2-P_(y^2)P_(w^2) |})

o24 : Sequence

i25 : ncBasis(2,D)

o25 = | x2 yx y2 zx zy z2 wx wy wz w2 ax ay az aw a2 |

              1      15
o25 : Matrix D  <-- D

i26 : isNormal (x^2+z^2-y^2-w^2)

o26 = true

i27 : E' = QQ[x,y,z,w,SkewCommutative=>true]

o27 = E'

o27 : PolynomialRing, 4 skew commutative variable(s)

i28 : E = toFreeAlgebraQuotient E'
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.

o28 = E

o28 : FreeAlgebraQuotient

i29 : f = map(E,C,gens E)

o29 = map (E, C, {x, y, z, w})

o29 : RingMap E <-- C

i30 : use C

o30 = C

o30 : FreeAlgebraQuotient

i31 : f x^2

o31 = 0

o31 : E

i32 : use E

o32 = E

o32 : FreeAlgebraQuotient

i33 : x^2 == 0

o33 = true

i34 : 
