2.3.4 Change of rings
---------------------

To calculate the local Milnor number we have to do the calculation with the
same commands in a ring with local ordering.
Define the localization of the polynomial ring at the origin
(see Polynomial data, and Mathematical background).

ring rl = 0,(x,y,z),ds;

This ordering determines the standard basis which will be calculated.
Fetch the polynomial defined in the ring r into this new ring,
thus avoiding retyping the input.

poly f = fetch(r,f);
f;
==> z2+x3+y3+x3y2-x2y3

Instead of fetch we can use the function imap
which is more general but less efficient.
The most general way to fetch data from one ring to another is to use maps,
this will be explained in map.

In this ring the terms are ordered by increasing exponents. The local Milnor
number is now

Milnor(f);
==> 4

This shows that f has outside the origin in affine 3-space
singularities with local Milnor number adding up to
$12-4=8$.
Using global and local orderings as above is a convenient way to check
whether a variety has singularities outside the origin.

The command jacob applied twice gives the Hessian of f, a
3x3 - matrix.

matrix H = jacob(jacob(f));
H;
==> H[1,1]=6x+6xy2-2y3
==> H[1,2]=6x2y-6xy2
==> H[1,3]=0
==> H[2,1]=6x2y-6xy2
==> H[2,2]=6y+2x3-6x2y
==> H[2,3]=0
==> H[3,1]=0
==> H[3,2]=0
==> H[3,3]=2

The print command displays the matrix in a nicer form.

print(H);
==> 6x+6xy2-2y3,6x2y-6xy2,  0,
==> 6x2y-6xy2,  6y+2x3-6x2y,0,
==> 0,          0,          2

We may calculate the determinant and (the ideal generated by all) minors of
a given size.

det(H);
==> 72xy+24x4-72x3y+72xy3-24y4-48x4y2+64x3y3-48x2y4
minor(H,1);  // the 1x1 - minors
==> _[1]=2
==> _[2]=6y+2x3-6x2y
==> _[3]=6x2y-6xy2
==> _[4]=6x2y-6xy2
==> _[5]=6x+6xy2-2y3

The algorithm of the standard basis computations may be
affected by the command option. For example, a reduced standard
basis of the ideal generated by the
$1 \times 1$-minors
 of H  is obtained in the following way:
option(redSB);
groebner(minor(H,1));
==> _[1]=1

This shows that 1 is contained in the ideal of the
$1 \times 1$-minors,
hence the corresponding variety is empty.


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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; User manual for <a href="http://www.singular.uni-kl.de/"><i>Singular</i></a> version 2-0-4, October 2002,
generated by <a href="http://www.gnu.org/software/texinfo/"><i>texi2html</i></a>.
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