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On the complexity of Milnor number computation for ICIS
https://www.singular.uni-kl.de/forum/viewtopic.php?f=10&t=1926		 Page 1 of 1
Author:  Dmitry [ Mon May 16, 2011 1:41 pm ]
Post subject:  On the complexity of Milnor number computation for ICIS
I was doing some particular examples and came across the following jumps in time elapsed to compute the Milnor number for an ICIS.

Let V_{p,q}\subset C^5 be the germ defined by two homogeneous polynomials, of degrees p and q. Suppose the polynomials are mutually generic, so that the projectivized tangent cone of their intersection is smooth. So, I get an ICIS of the simplest type.

I tried to compute their Milnor numbers (Glory to Singular!) in some particular cases. In most cases the computation is done in less than a second. e.g. for (p,q)=(4,8) or (p,q)=(5,6). But in some simpler(!!!) cases Singular is stuck (I shut it down after several minutes of silence). For example, for (p,q)=(3,5) or (p,q)=(4,7). Naively I would expect the time elapsed to increase monotonically in (p,q). Any explanation, advice of how to choose (p,q) for fast time computing?

My (trivial) code:
LIB "sing.lib";
ring r = 0,(x,y,z,w,t),ds;
int p=3;int q=5;
ideal j=x^p+y^p+z^p+w^p+2099t^p,3x^q+2y^q+7z^q+1093w^q-1091t^q;milnor(j);

ps. Here 2099, 1093, 1091 are some big prime numbers, I tried to substitute in hope to speed up the process.
Author:  gorzel [ Mon May 16, 2011 3:01 pm ]
Post subject:  Re: On the complexity of Milnor number computation for ICIS
The general hint, if you want to compute Minor numbers fast,
is to do it over larger prime characteristic.

So instead of choosing general prime numbers for the coefficients
define for instance:

Quote:
ring r32003,(x,y,z,w,t),ds;
int p=3;int q=5;
ideal j=x^p+y^p+z^p+w^p+2099t^p,3x^q+2y^q+7z^q+1093w^q-1091t^q;
option(prot);
milnor(j);


Not that the prime charcteristic is should be so large such that
coefficients of jacob(j), or at least the degrees of the terms of f
are relatively prime (gcd == 1) to the characteristic.

After this you may recheck it in char 0. The output of option(prot) in char > 0
shows you at which stage you are in char 0 with the computation.
Author:  Dmitry [ Mon May 16, 2011 6:10 pm ]
Post subject:  Re: On the complexity of Milnor number computation for ICIS
Thanks! :D
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