Singular Manual: arnoldClassify

| #### D.15.1.4 arnoldClassify Procedure from library `arnoldclassify.lib` (see arnoldclassify_lib). Usage: arnoldClassify (f); f poly Assume: The basering is local of characteristic 0 and f defines an isolated singularity from Arnol'd's list of corank at most 2. Compute: singularity class with respect to right equivalence and invariants used in the process of classification Return: Singularity class of f of type singclass containing - name of singularity series as listed by arnoldListAllSeries(), - name of singularity class, - parameters k,r,s defining the singularity class, -1 if not used, - modality, corank, Milnor number, determinacy, - Tjurina number, -2 if not computed, -1 if infinite, - Milnor code, -1 if not computed, - normal form of the singularity series from Arnol'd's list, - restrictions on parameters as string in SINGULAR syntax. Example:
    LIB "arnoldclassify.lib";
    ring r = 0,(x,y),ds;
    int k = random(3,10);
    poly g = x4 + x2*y^(2*k+1)+x*y^(3*k+1)+ y^(4*k +1);
    arnoldClassify(g);
    ==> Series=W[12k]
    ==> Class=W[60]
    ==> k=5
    ==> r=-1
    ==> s=-1
    ==> Modality=13
    ==> Corank=2
    ==> Milnor=60
    ==> Determinacy=29
    ==> Tjurina=-2
    ==> MilnorCode=1,1,10,9,9
    ==> NormalForm=x4+a(y)*x*y^(3*k+1)+c(y)*x^2*y^(2*k+1)+y^(4*k+1)
    ==> Restrictions=(k>=1)&&(k>1||a==0)&&(deg(a)<=(k-2))&&
    ==>         (deg(c)<=(2*k-2))
    map phi=r,y-x^2,x+y;
    phi(g);
    ==> y4-4x2y3+6x4y2-4x6y+x8+x11y2+11x10y3+55x9y4+165x8y5+330x7y6+462x6y7+462x5\
       y8+330x4y9+165x3y10+55x2y11+11xy12+y13-2x13y-22x12y2-110x11y3-330x10y4-66\
       0x9y5-924x8y6-924x7y7-660x6y8-330x5y9-110x4y10-22x3y11-2x2y12+x15+11x14y+\
       55x13y2+165x12y3+330x11y4+462x10y5+462x9y6+330x8y7+165x7y8+55x6y9+11x5y10\
       +x4y11+x16y+16x15y2+120x14y3+560x13y4+1820x12y5+4368x11y6+8008x10y7+11440\
       x9y8+12870x8y9+11440x7y10+8008x6y11+4368x5y12+1820x4y13+560x3y14+120x2y15\
       +16xy16+y17-x18-16x17y-120x16y2-560x15y3-1820x14y4-4368x13y5-8008x12y6-11\
       440x11y7-12870x10y8-11440x9y9-8008x8y10-4368x7y11-1820x6y12-560x5y13-120x\
       4y14-16x3y15-x2y16+x21+21x20y+210x19y2+1330x18y3+5985x17y4+20349x16y5+542\
       64x15y6+116280x14y7+203490x13y8+293930x12y9+352716x11y10+352716x10y11+293\
       930x9y12+203490x8y13+116280x7y14+54264x6y15+20349x5y16+5985x4y17+1330x3y1\
       8+210x2y19+21xy20+y21
    arnoldClassify(phi(g));
    ==> Series=W[12k]
    ==> Class=W[60]
    ==> k=5
    ==> r=-1
    ==> s=-1
    ==> Modality=13
    ==> Corank=2
    ==> Milnor=60
    ==> Determinacy=29
    ==> Tjurina=-2
    ==> MilnorCode=1,1,10,9,9
    ==> NormalForm=x4+a(y)*x*y^(3*k+1)+c(y)*x^2*y^(2*k+1)+y^(4*k+1)
    ==> Restrictions=(k>=1)&&(k>1||a==0)&&(deg(a)<=(k-2))&&
    ==>         (deg(c)<=(2*k-2))
    ring C = (0,i), (x,y), ds;
    minpoly = i2 + 1;
    poly f =(x2+y2)^2+x5;
    arnoldClassify(f);
    ==> Series=Y[1,r,s]
    ==> Class=Y[1,1,1]
    ==> k=1
    ==> r=1
    ==> s=1
    ==> Modality=1
    ==> Corank=2
    ==> Milnor=11
    ==> Determinacy=5
    ==> Tjurina=-2
    ==> MilnorCode=1,1,1,2,1
    ==> NormalForm= x^(4+r)+ a(y)*x2*y2 + y^(4+s)
    ==> Restrictions=(deg(a)==0)&&(jet(a,0)!=0)&&(1<=s)&&(s<=r)

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