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D.6.4.2 classify

Procedure from library classify.lib (see classify_lib).

Usage:
classify(f); f=poly

Compute:
normal form and singularity type of f with respect to right equivalence, as given in the book "Singularities of differentiable maps, Volume I" by V.I. Arnold, S.M. Gusein-Zade, A.N. Varchenko

Return:
normal form of f, of type poly

Remark:
This version of classify is only beta. Please send bugs and comments to: "Kai Krueger" <krueger@mathematik.uni-kl.de>
Be sure to have at least Singular version 1.0.1.

Note:
type init_debug(n); (0 <= n <= 10) in order to get intermediate information, higher values of n give more information.
The proc creates several global objects with names all starting with @, hence there should be no name conflicts.

Example:
 
LIB "classify.lib";
ring r=0,(x,y,z),ds;
poly f=(x2+3y-2z)^2+xyz-(x-y3+x2*z3)^3;
classify(f);
==> About the singularity :
==>           Milnor number(f)   = 4
==>           Corank(f)          = 2
==>           Determinacy       <= 5
==> Guessing type via Milnorcode:   D[k]=D[4]
==> 
==> Computing normal form ...
==> I have to apply the splitting lemma. This will take some time....:-)
==>    Arnold step number 4
==> The singularity
==>    -x3+3/2xy2+1/2x3y-1/16x2y2+3x2y3
==> is R-equivalent to D[4].
==>    Milnor number = 4
==>    modality      = 0
==> 2z2+x2y+y3
init_debug(3);
==> Debugging level change from  0  to  3
classify(f);
==> Computing Basicinvariants of f ...
==> About the singularity :
==>           Milnor number(f)   = 4
==>           Corank(f)          = 2
==>           Determinacy       <= 5
==> Hcode: 1,2,1,0,0
==> Milnor code :  1,1,1
==> Debug:(2):  entering HKclass3_teil_1 1,1,1
==> Debug:(2):  finishing HKclass3_teil_1
==> Guessing type via Milnorcode:   D[k]=D[4]
==> 
==> Computing normal form ...
==> I have to apply the splitting lemma. This will take some time....:-)
==> Debug:(3):  Split the polynomial below using determinacy:  5
==> Debug:(3):  9y2-12yz+4z2-x3+6x2y-4x2z+xyz+x4+3x2y3
==> Debug:(2):  Permutations: 3,2,1
==> Debug:(2):  Permutations: 3,2,1
==> Debug:(2):  rank determined with Morse rg= 1
==> Residual singularity f= -x3+3/2xy2+1/2x3y-1/16x2y2+3x2y3
==> Step 3
==>    Arnold step number 4
==> The singularity
==>    -x3+3/2xy2+1/2x3y-1/16x2y2+3x2y3
==> is R-equivalent to D[4].
==>    Milnor number = 4
==>    modality      = 0
==> Debug:(2):  Decode:
==> Debug:(2):  S_in= D[4]   s_in= D[4]                          
==> Debug:(2):  Looking for Normalform of  D[k] with (k,r,s) = ( 4 , 0 , 0 )
==> Debug:(2):  Opening Singalarity-database:  
==>  DBM: NFlist
==> Debug:(2):  DBMread( D[k] )= x2y+y^(k-1) .
==> Debug:(2):  S= f = x2y+y^(k-1);  Tp= x2y+y^(k-1) Key= I_D[k]
==> Polynom f= x2y+y3   crk= 2   Mu= 4  MlnCd= 1,1,1
==> Debug:(2):  Info= x2y+y3
==> Debug:(2):  Normal form NF(f)= 2*x(3)^2+x(1)^2*x(2)+x(2)^3
==> 2z2+x2y+y3


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