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PACKAGE : resol2.mu
CONTENTS : Functions to compute the resolution of singularities
of a singular 1-form in the complex plane.
AUTHOR : P. Fortuny (pfortuny@agt.uva.es)
STATUS : Experimental, although almost completely functional
SYSTEMS : Any with MuPAD
INSTALLED : 21.01.2000
What is it for?
---------------
(Some knowledge of resolution of singularities is assumed).
Give a polynomial plane vector field at (0,0), there is a
finite sequence of blowing-ups which transforms the singularity
into so-called "simple" one, which are those with non-nilpo-
tent linear part and such that the eigenvalues of this linear
part have not a positive rational quotient.
The process has some associated invariants: the dual graph of
the reduction and the quotient of eigenvalues at the final
singularities. These are computed by RESOL2. Moreover, it
also calculates the separatrices (power series expansions of
the solutions) up to a (up to date fixed, but easily modifiable)
given degree.
There are (undocumented still) functions to produce PS-draw
files to draw the graph and place it in a LaTeX document. See
the comments at the beginning of each .mu file.
How to install it?
------------------
Place the .tar archive in the same directory of the MuPAD
library (unix systems ...MuPAD/share/lib).
How to use it?
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Export the library
>> export(resol2);
and use the functions. The easiest way to reduce a 1-form
a(x,y) dx + b(x,y) dy
is to write
>> red([a,b]);
(notice that the variables in a and b MUST be 'x' and 'y';
there are other ways to use different variables, but they
require more information than what this info-file is for)
the output is something like
[[[a1,b1],[a2,b2],...,[ak,bk]],k], Network(), table(1=table(
1=f(t),2=g(t),3="str")
which means
ai-> kind of singularity (corner or regular point)
bi-> quotient of the eigenvalues
Network-> dual graph of the reduction
table()....-> parametrization of the separatrices at
the regular points
>> setuserinfo(resol2,1) gives information on the reduction
process (which blowing-ups are done and where)
>> setuserinfo(resol2,2) even more.
There is more information in the .tar archive, in the files
RESOL2/DOC/resol2.help
RESOL2/resol2.txt
and at the beginning of the .mu files in
RESOL2/RESOL2
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