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Intersection Multiplicity of Hypersurfaces http://www.singular.unikl.de/forum/viewtopic.php?f=10&t=2420 
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Author:  Vrbik [ Fri Jun 20, 2014 10:14 pm ] 
Post subject:  Intersection Multiplicity of Hypersurfaces 
I am a Singular newbie. Suppose the ambient polynomial ring is QQ[x0,...,xN] for QQ a field. How would one calculate the intersection multiplicity of a point in the variety of N curves h1,...,hN in QQ[x1,...,xN] at a point from Q x ... x Q. This value is equivalent to the (vector) dimension of the local ring at p modulo <h1,...,hN>. In particular, im( p, h1,...,hN ) = dim( {f/g : g(p) <> 0} / <h1,...,hN> ); 
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