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the number of irreducible analytic components at a point
https://www.singular.uni-kl.de/forum/viewtopic.php?f=10&t=1701		 Page 1 of 1
Author:  tortoisesaid [ Mon Feb 02, 2009 4:54 am ]
Post subject:  the number of irreducible analytic components at a point
Hello all:

I have a geometry question. Given an ALGEBRAIC variety V in C^n
defined by a single polynomial and given a point p in V, can Singular compute the number of (distinct) irreducible ANALYTIC components of V
passing through p?

For example, let f = y^2 -x^2*(1 +x). Then the variety V(f) has two
irreducible analytic components passing through (0,0), one for each
factor of the decomposition

f = (y -x*sqrt(1+x)) *(y +x*sqrt(1+x))

in C{x,y}, the ring of power series convergent in a neighborhood of
(0,0).

I was told by someone on the Sage mailing list that Singular can factor approximately (up to any given degree) in C[[x_1,...,x_n]], the ring of formal power series over C. If so, what are the appropriate commands? (I tried searching the Singular documentation but couldn't find an answer.) Using such a factorization would be one way to determine the number of irreducible analytic components at p.

Thanks for your attention.
Alex
Author:  anne [ Mon Feb 02, 2009 9:45 pm ]
Post subject:  Re: number of irred analytic components at a point
If n=2 and p is (0,0), this is indeed available in Singular by the following
sequence of commands:

LIB "hnoether.lib";
ring r=0,(x,y),ds;
poly f=y^2 -x^2*(1 +x);
list HNoetherExpansion=hnexpansion(f);
displayHNE(HNoetherExpansion);

As output you obtain:
// Hamburger-Noether development of branch nr.1:
y = x+1/2*x^2 + ..... (terms of degree >=3)

// Hamburger-Noether development of branch nr.2:
y = -x-1/2*x^2 + ..... (terms of degree >=3)

If p is not the origin, you need to do a coordinate change first to move to (0,0).

If n>2, then Hamburger-Noether Expansion can no longer be used and there is
no application ready tool in this situation in SINGULAR.

Do you need it for n=2 or n>2?
Author:  tortoisesaid [ Mon Feb 02, 2009 11:06 pm ]
Post subject: 
Hi Anne:

Thanks for your help. I need a test for n >= 2. Is there a way is Singular to factor polynomials up to a given degree in C[[x_1,...,x_n]]?
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