normalElements(RingMap,ZZ) -- Finds elements normalized by a ring map

Synopsis

Function: normalElements
Usage:

normalElements(f,n)
Inputs:
- f, a ring map,
- n, an integer, a homogeneous degree in which to search for normal elements
Outputs:
- a matrix,

Description

A normal element x in a non-commutative ring R determines an automorphism f of R by a*x=x*f(a). Conversely, given a ring endomorphism, we may ask if any x satisfy the above equation for all a.

Given a ring map f and a degree n, this method returns solutions to the equations a*x=x*f(a) for all generators a of R.

i1 : B = skewPolynomialRing(QQ,(-1)_QQ,{x,y,z,w})
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.

o1 = B

o1 : FreeAlgebraQuotient

i2 : sigma = map(B,B,{y,z,w,x})

o2 = map (B, B, {y, z, w, x})

o2 : RingMap B <-- B

i3 : C = oreExtension(B,sigma,a)
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.

o3 = C

o3 : FreeAlgebraQuotient

i4 : sigmaC = map(C,C,{y,z,w,x,a})

o4 = map (C, C, {y, z, w, x, a})

o4 : RingMap C <-- C

i5 : normalElements(sigmaC,1)

o5 = | a |

             1      1
o5 : Matrix C  <-- C

i6 : normalElements(sigmaC,2)

o6 = 0

              1
o6 : Matrix QQ  <-- 0

i7 : normalElements(sigmaC * sigmaC,2)

o7 = | a2 |

             1      1
o7 : Matrix C  <-- C

i8 : normalElements(sigmaC * sigmaC * sigmaC, 3)

o8 = | a3 |

             1      1
o8 : Matrix C  <-- C

Ways to use this method: