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kad
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Post subject: milnor and tjurina numbers Posted: Wed Oct 04, 2006 11:39 am |
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Bonjour,
can you me say why the procedure milnor and tjurina gives the same numbers for
{x^{2} + y^{3} + z^{3} +zy=0} or
{x^{2} + y^{3} + y^{4} +yz^{2}=0} ?
thanks. kaddar.
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greuel
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Post subject: Posted: Sat Dec 09, 2006 1:53 am |
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Joined: Mon Aug 29, 2005 9:22 am Posts: 41 Location: Kaiserslautern, Germany
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Please provide the full Singular input for your examples.
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greuel
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Post subject: Posted: Thu Jan 25, 2007 5:25 pm |
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Joined: Mon Aug 29, 2005 9:22 am Posts: 41 Location: Kaiserslautern, Germany
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I recomputed the examples: LIB"sing.lib"; ring r = 0,(x,y,z),ds; poly f = x2 + y3 + z3 +zy; poly g = x2 + y3 + y4 +yz2; milnor(f); tjurina(f); //1 milnor(g); tjurina(g); //4
The answer is correct since f is contained in the ideal of the partials. f is an A_1 and g a D_4 singularity. You get this by applying classify from classify.lib.
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renzo
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Post subject: Posted: Sat Feb 03, 2007 1:30 am |
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Joined: Sat Feb 03, 2007 1:14 am Posts: 1 Location: Albuquerque, USA
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I think it is worth mentioning that by a theorem of Saito any weighted homogeneous polynomial defining an IHS belongs to its Jacobi ideal and viceversa, therefore they have the same Turina and Milnor numbers.
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