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7.5.12.0. ncfactor
Procedure from library ncfactor.lib (see ncfactor_lib).
- Usage:
- ncfactor(h); h is a polynomial in a non-commutative polynomial
algebra over a field k.
- Return:
- list(list)
- Purpose:
- Compute all factorizations of h.
- Theory:
- Implements an ansatz-driven factorization method as outlined
by Bell, Heinle and Levandovskyy in "On Noncommutative Finite
Factorization Domains".
- Assume:
- - k is a ring, such that factorize can factor any univariate and
multivariate commutative polynomial over k.
- There exists at least one variable in the ring.
- Note:
- - works for both PLURAL and LETTERPLACE subsystems
- Every entry of the output list is a list with factors for one possible factorization.
The first factor is always a constant (1, if no nontrivial constant
could be excluded).
Example:
| LIB "ncfactor.lib";
// first, an example with PLURAL
def R = makeUsl2();
setring(R);
poly p = e^3*f+e^2*f^2-e^3+e^2*f+2*e*f^2-3*e^2*h-2*e*f*h-8*e^2
+e*f+f^2-4*e*h-2*f*h-7*e+f-h;
ncfactor(p);
==> [1]:
==> [1]:
==> 1
==> [2]:
==> e+1
==> [3]:
==> ef-e+f-2h-3
==> [4]:
==> e+f
==> [2]:
==> [1]:
==> 1
==> [2]:
==> e2f+ef2-e2+f2-2eh-3e-f-2h
==> [3]:
==> e+1
kill R;
// an example with LETTERPLACE
LIB "freegb.lib";
==> // ** redefining tstfreegb (LIB "freegb.lib";) ./examples/ncfactor.sing:1\
0
==> // ** redefining tstfreegb (LIB "freegb.lib";) ./examples/ncfactor.sing:1\
0
==> // ** redefining setLetterplaceAttributes (LIB "freegb.lib";) ./examples/\
ncfactor.sing:10
==> // ** redefining setLetterplaceAttributes (LIB "freegb.lib";) ./examples/\
ncfactor.sing:10
==> // ** redefining lst2str (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining mod2str (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining vct2str (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining isVar (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining letplaceGBasis (LIB "freegb.lib";) ./examples/ncfactor.s\
ing:10
==> // ** redefining letplaceGBasis (LIB "freegb.lib";) ./examples/ncfactor.s\
ing:10
==> // ** redefining lieBracket (LIB "freegb.lib";) ./examples/ncfactor.sing:\
10
==> // ** redefining lieBracket (LIB "freegb.lib";) ./examples/ncfactor.sing:\
10
==> // ** redefining lpPrint (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining lpPrint (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining freeGBasis (LIB "freegb.lib";) ./examples/ncfactor.sing:\
10
==> // ** redefining freeGBasis (LIB "freegb.lib";) ./examples/ncfactor.sing:\
10
==> // ** redefining crs (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining polylen (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining lpDegBound (LIB "freegb.lib";) ./examples/ncfactor.sing:\
10
==> // ** redefining lpDegBound (LIB "freegb.lib";) ./examples/ncfactor.sing:\
10
==> // ** redefining lpVarBlockSize (LIB "freegb.lib";) ./examples/ncfactor.s\
ing:10
==> // ** redefining lpVarBlockSize (LIB "freegb.lib";) ./examples/ncfactor.s\
ing:10
==> // ** redefining isFreeAlgebra (LIB "freegb.lib";) ./examples/ncfactor.si\
ng:10
==> // ** redefining isFreeAlgebra (LIB "freegb.lib";) ./examples/ncfactor.si\
ng:10
==> // ** redefining lpNcgenCount (LIB "freegb.lib";) ./examples/ncfactor.sin\
g:10
==> // ** redefining lpNcgenCount (LIB "freegb.lib";) ./examples/ncfactor.sin\
g:10
==> // ** redefining makeLetterplaceRing (LIB "freegb.lib";) ./examples/ncfac\
tor.sing:10
==> // ** redefining makeLetterplaceRing (LIB "freegb.lib";) ./examples/ncfac\
tor.sing:10
==> // ** redefining makeLetterplaceRing1 (LIB "freegb.lib";) ./examples/ncfa\
ctor.sing:10
==> // ** redefining makeLetterplaceRing2 (LIB "freegb.lib";) ./examples/ncfa\
ctor.sing:10
==> // ** redefining makeLetterplaceRing4 (LIB "freegb.lib";) ./examples/ncfa\
ctor.sing:10
==> // ** redefining makeLetterplaceRing3 (LIB "freegb.lib";) ./examples/ncfa\
ctor.sing:10
==> // ** redefining freegbold (LIB "freegb.lib";) ./examples/ncfactor.sing:1\
0
==> // ** redefining stringpoly2lplace (LIB "freegb.lib";) ./examples/ncfacto\
r.sing:10
==> // ** redefining addplaces (LIB "freegb.lib";) ./examples/ncfactor.sing:1\
0
==> // ** redefining sent2lplace (LIB "freegb.lib";) ./examples/ncfactor.sing\
:10
==> // ** redefining testnumber (LIB "freegb.lib";) ./examples/ncfactor.sing:\
10
==> // ** redefining str2lplace (LIB "freegb.lib";) ./examples/ncfactor.sing:\
10
==> // ** redefining strpower2rep (LIB "freegb.lib";) ./examples/ncfactor.sin\
g:10
==> // ** redefining shiftPoly (LIB "freegb.lib";) ./examples/ncfactor.sing:1\
0
==> // ** redefining lastBlock (LIB "freegb.lib";) ./examples/ncfactor.sing:1\
0
==> // ** redefining test_shift (LIB "freegb.lib";) ./examples/ncfactor.sing:\
10
==> // ** redefining lp2lstr (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining lp2lstr (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining strList2poly (LIB "freegb.lib";) ./examples/ncfactor.sin\
g:10
==> // ** redefining file2lplace (LIB "freegb.lib";) ./examples/ncfactor.sing\
:10
==> // ** redefining lpPower (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining lpNF (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining lpNF (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining lpDivision (LIB "freegb.lib";) ./examples/ncfactor.sing:\
10
==> // ** redefining lpDivision (LIB "freegb.lib";) ./examples/ncfactor.sing:\
10
==> // ** redefining lpGBPres2Poly (LIB "freegb.lib";) ./examples/ncfactor.si\
ng:10
==> // ** redefining lpGBPres2Poly (LIB "freegb.lib";) ./examples/ncfactor.si\
ng:10
==> // ** redefining getExpVecs (LIB "freegb.lib";) ./examples/ncfactor.sing:\
10
==> // ** redefining delSupZero (LIB "freegb.lib";) ./examples/ncfactor.sing:\
10
==> // ** redefining delSupZeroList (LIB "freegb.lib";) ./examples/ncfactor.s\
ing:10
==> // ** redefining makeDVec (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining makeDVecL (LIB "freegb.lib";) ./examples/ncfactor.sing:1\
0
==> // ** redefining makeDVecI (LIB "freegb.lib";) ./examples/ncfactor.sing:1\
0
==> // ** redefining dShiftDiv (LIB "freegb.lib";) ./examples/ncfactor.sing:1\
0
==> // ** redefining lpNormalForm1 (LIB "freegb.lib";) ./examples/ncfactor.si\
ng:10
==> // ** redefining lpNormalForm2 (LIB "freegb.lib";) ./examples/ncfactor.si\
ng:10
==> // ** redefining isOrderingShiftInvariant (LIB "freegb.lib";) ./examples/\
ncfactor.sing:10
==> // ** redefining isOrderingShiftInvariant (LIB "freegb.lib";) ./examples/\
ncfactor.sing:10
==> // ** redefining lpMonomialsWithHoles (LIB "freegb.lib";) ./examples/ncfa\
ctor.sing:10
==> // ** redefining getlpCoeffs (LIB "freegb.lib";) ./examples/ncfactor.sing\
:10
==> // ** redefining lpReduce (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining entryViolation (LIB "freegb.lib";) ./examples/ncfactor.s\
ing:10
==> // ** redefining checkAssumptionsLPIV (LIB "freegb.lib";) ./examples/ncfa\
ctor.sing:10
==> // ** redefining checkAssumptions (LIB "freegb.lib";) ./examples/ncfactor\
.sing:10
==> // ** redefining checkLPRing (LIB "freegb.lib";) ./examples/ncfactor.sing\
:10
==> // ** redefining checkAssumptionIdeal (LIB "freegb.lib";) ./examples/ncfa\
ctor.sing:10
==> // ** redefining checkAssumptionPoly (LIB "freegb.lib";) ./examples/ncfac\
tor.sing:10
==> // ** redefining isContainedInVp (LIB "freegb.lib";) ./examples/ncfactor.\
sing:10
==> // ** redefining extractLinearPart (LIB "freegb.lib";) ./examples/ncfacto\
r.sing:10
==> // ** redefining isLinearVector (LIB "freegb.lib";) ./examples/ncfactor.s\
ing:10
==> // ** redefining lpAssumeViolation (LIB "freegb.lib";) ./examples/ncfacto\
r.sing:10
==> // ** redefining skip0 (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining ivL2lpI (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining ivL2lpI (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining iv2lp (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining iv2lp (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining iv2lpList (LIB "freegb.lib";) ./examples/ncfactor.sing:1\
0
==> // ** redefining iv2lpList (LIB "freegb.lib";) ./examples/ncfactor.sing:1\
0
==> // ** redefining iv2lpMat (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining iv2lpMat (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining lpId2ivLi (LIB "freegb.lib";) ./examples/ncfactor.sing:1\
0
==> // ** redefining lpId2ivLi (LIB "freegb.lib";) ./examples/ncfactor.sing:1\
0
==> // ** redefining lp2iv (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining lp2iv (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining lp2ivId (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining lp2ivId (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining testLift (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining testLift (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining testSyz (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining testSyz (LIB "freegb.lib";) ./examples/ncfactor.sing:10
==> // ** redefining mod_init (LIB "freegb.lib";) ./examples/ncfactor.sing:10
ring r = 0,(x,y),Dp;
def R = freeAlgebra(r,5); setring(R);
poly p = x*y*x - x;
ncfactor(p);
==> [1]:
==> [1]:
==> 1
==> [2]:
==> x*y-1
==> [3]:
==> x
==> [2]:
==> [1]:
==> 1
==> [2]:
==> x
==> [3]:
==> y*x-1
| See also:
facSubWeyl;
facWeyl;
testNCfac.
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