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

Synopsis

• Usage:
multiplicity I
multiplicity(I,f)
• Inputs:
• I, an ideal
• f, , optional argument, if given it should be a non-zero divisor in the ideal I
• Optional inputs:
• Outputs:
• an integer, the normalized leading coefficient of the Hilbert-Samuel polynomial of I

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

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

Ways to use multiplicity :

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