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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" <[email protected]>
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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