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ReesAlgebra :: multiplicity

multiplicity -- Compute the Hilbert-Samuel multiplicity of an ideal



Given an ideal I⊂ R, “multiplicity I” returns the degree of the normal cone of I. When R/I has finite length this is the sum of the Samuel multiplicities of I at the various localizations of R. When I is generated by a complete intersection, this is the length of the ring R/I but in general it is greater. For example,

i1 : R=ZZ/101[x,y]

o1 = R

o1 : PolynomialRing
i2 : I = ideal(x^3, x^2*y, y^3)

             3   2    3
o2 = ideal (x , x y, y )

o2 : Ideal of R
i3 : multiplicity I

o3 = 9
i4 : degree I

o4 = 7


The normal cone is computed using the Rees algebra, thus may be slow.

Ways to use multiplicity :