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A.4.1 Milnor and Tjurina number
The Milnor number, resp. the Tjurina number, of a power
series f in
is
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
the localization of
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
prot to have a short protocol during standard basis
computation
-
define the ring
r1 of characteristic 32003 with variables x,y,z ,
monomial ordering ds , series ring (i.e., K[x,y,z] localized at (x,y,z))
-
list the information about
r1 by typing its name
-
define the integers
a,b,c,t
-
define a polynomial
f (depending on a,b,c,t ) and display it
-
define the jacobian ideal
i of f
-
compute a standard basis of
i
-
compute the Milnor number (=250) with
vdim and 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
See also sing_lib for the library commands for the computation of
the Milnor and Tjurina number.
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