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.