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A.1 Milnor and TjurinaThe Milnor number, resp. the Tjurina number, of a power
series f in
respectively where jacob(f) is the ideal generated by the partials
of f. tjurina(f) is finite, if and only if f has an
isolated singularity. The same holds for milnor(f) if
K has characteristic 0.
SINGULAR displays -1 if the dimension is infinite.
SINGULAR cannot compute with infinite power series. But it can
work in
We shall show in the example below how to realize the following:
option(prot);
ring r1 = 32003,(x,y,z),ds;
r1;
→ // characteristic : 32003
→ // number of vars : 3
→ // block 1 : ordering ds
→ // : names x y z
→ // block 2 : ordering C
int a,b,c,t=11,5,3,0;
poly f = x^a+y^b+z^(3*c)+x^(c+2)*y^(c-1)+x^(c-1)*y^(c-1)*z3+
x^(c-2)*y^c*(y^2+t*x)^2;
f;
→ y5+x5y2+x2y2z3+xy7+z9+x11
ideal i=jacob(f);
i;
→ i[1]=5x4y2+2xy2z3+y7+11x10
→ i[2]=5y4+2x5y+2x2yz3+7xy6
→ i[3]=3x2y2z2+9z8
ideal j=std(i);
→ [1023:2]7(2)s8s10s11s12s(3)s13(4)s(5)s14(6)s(7)15--.s(6)-16.-.s(5)17.s(7)\
s--s18(6).--19-..sH(24)20(3)...21....22....23.--24-
→ product criterion:10 chain criterion:69
"The Milnor number of f(11,5,3) for t=0 is", vdim(j);
→ The Milnor number of f(11,5,3) for t=0 is 250
j=i+f; // overwrite j
j=std(j);
→ [1023:2]7(3)s8(2)s10s11(3)ss12(4)s(5)s13(6)s(8)s14(9).s(10).15--sH(23)(8)\
...16......17.......sH(21)(9)sH(20)16(10).17...........18.......19..----.\
.sH(19)
→ product criterion:10 chain criterion:53
vdim(j); // compute the Tjurina number for t=0
→ 195
t=1;
f=x^a+y^b+z^(3*c)+x^(c+2)*y^(c-1)+x^(c-1)*y^(c-1)*z3
+x^(c-2)*y^c*(y^2+t*x)^2;
ideal i1=jacob(f);
ideal j1=std(i1);
→ [1023:2]7(2)s8s10s11s12s13(3)ss(4)s14(5)s(6)s15(7).....s(8)16.s...s(9)..1\
7............s18(10).....s(11)..-.19.......sH(24)(10).....20...........21\
..........22.............................23..............................\
.24.----------.25.26
→ product criterion:11 chain criterion:83
"The Milnor number of f(11,5,3) for t=1:",vdim(j1);
→ The Milnor number of f(11,5,3) for t=1: 248
vdim(std(j1+f)); // compute the Tjurina number for t=1
→ [1023:2]7(16)s8(15)s10s11ss(16)-12.s-s13s(17)s(18)s(19)-s(18).-14-s(17)-s\
(16)ss(17)s15(18)..-s...--.16....-.......s(16).sH(23)s(18)...17..........\
18.........sH(20)17(17)....................18..........19..---....-.-....\
.....20.-----...s17(9).........18..............19..-.......20.-......21..\
.......sH(19)16(5).....18......19.-----
→ product criterion:15 chain criterion:174
→ 195
option(noprot);
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