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A.4.1 Milnor and Tjurina number
The Milnor number, resp. the Tjurina number, of a power
series f in
![$K[[x_1,\ldots,x_n]]$](sing_491.png){kind=small}
is
![\begin{displaymath}
\hbox{milnor}(f) = \hbox{dim}_K(K[[x_1,\ldots,x_n]]/\hbox{jacob}(f)),
\end{displaymath}](sing_492.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_493.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_494.png){kind=small}
,the localization of
![$K[x_1,\ldots,x_n]$](sing_62.png){kind=small}
at the maximal ideal
.To do this, one has to define a ring with a local monomial ordering
such as ds, Ds, ls, ws, Ws (the second letter 's' referring to power
's'eries), or an appropriate matrix ordering.
See Monomial orderings for a menu of possible orderings.
For theoretical reasons, the vector space dimension computed over the localization ring coincides with the Milnor (resp. Tjurina) number as defined above (in the power series ring).
We show in the example below the following:
-
set option
protto have a short protocol during standard basis computation -
define the ring
r1of characteristic 32003 with 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;
==> // coefficients: ZZ/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);
==> 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; // override j
j=std(j);
==> 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)18(9)sH(20)(8)s17(10)..18..-.......19..---..sH(19)
==> product criterion:11 chain criterion:62
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);
==> 7(2)s8s10s11s12s13(3)ss(4)s14(5)s(6)s15(7).....s(8)16.s...s(9)..17.......\
.....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
==> 7(16)s8(15)s10s11ss(16)-12.s-s13s(17)s(18)s(19)-s(18).-14-s(17)-s(16)ss(1\
7)s15(18)..-s...--.16....-.......s(16).sH(23)s(18)...17..........18......\
...sH(20)17(17)18...19..--...-.-...-..........20.-----s17(9)..18...19..-2\
0.-.............21..........sH(19)(4)----
==> product criterion:15 chain criterion:174
==> 195
option(noprot);
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