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A.1 Milnor and Tjurina
The Milnor number, resp. the Tjurina number, of a power
series f in
![$K[[x_1,\ldots,x_n]]$](sing_150.png){kind=small}
is
![\begin{displaymath}
\hbox{milnor}(f) = \hbox{dim}_K(K[[x_1,\ldots,x_n]]/\hbox{jacob}(f)),
\end{displaymath}](sing_151.png){kind=small}
respectively
![\begin{displaymath}
\hbox{tjurina}(f) = \hbox{dim}_K(K[[x_1,\ldots,x_n]]/((f)+\hbox{jacob}(f)))
\end{displaymath}](sing_152.png){kind=small}
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
![$\hbox{Loc}_{(x)}K[x_1,\ldots,x_n]$](sing_153.png){kind=small}
,
the localization of
![$K[x_1,\ldots,x_n]$](sing_55.png){kind=small}
at the maximal ideal
.
To do this one has to define an
s-ordering like ds, Ds, ls, ws, Ws or an appropriate matrix
ordering (look at the manual to get information about the possible
monomial orderings in SINGULAR, or type help Monomial orderings;
to get a menu of possible orderings. For further help type, e.g.,
help local orderings;).
See Monomial orderings.
We shall show in the example below how to realize the following:
-
set option
protto have a short protocol during standard basis computation -
define the ring
r1with char 32003, variablesx,y,z, monomial orderingds, series ring (i.e., K[x,y,z] localized at (x,y,z)) -
list the information about
r1by typing its name -
define the integers
a,b,c,t -
define a polynomial
f(depending ona,b,c,t) and display it -
define the jacobian ideal
ioff -
compute a standard basis of
i -
compute the Milnor number (=250) with
vdimand create and display a string in order to comment the result (text between quotes " "; is a ’string’) -
compute a standard basis of
i+(f) -
compute the Tjurina number (=195) with
vdim -
then compute the Milnor number (=248) and the Tjurina number
(=195) for
t=1 -
reset the option to
noprot
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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