# Singular

#### 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) ```