Singular Manual: ncfactor

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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