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

Synopsis

Description

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

Caveat

Ways to use multiplicity :

  • multiplicity(Ideal)
  • multiplicity(Ideal,RingElement)