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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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