multiplicity -- This method computes the algebraic (Hilbert-Samuel) multiplicity

Synopsis

Usage:

multiplicity(IX,IY)
Inputs:
- IX, an ideal, a multi-homogeneous prime ideal defining a closed subscheme of \PP^{n_1}x...x\PP^{n_m}; makeProductRing builds the graded coordinate ring of \PP^{n_1}x...x\PP^{n_m}.
- IY, an ideal, a multi-homogeneous primary ideal defining a closed subscheme of \PP^{n_1}x...x\PP^{n_m}; makeProductRing builds the graded coordinate ring of \PP^{n_1}x...x\PP^{n_m}.
Optional inputs:
- Verbose (missing documentation) => a Boolean value, default value false,
Outputs:
- eXY, an integer, the algebraic (Hilbert-Samuel) multiplicity e_XY of the variety X associated to IX in the scheme Y associated to IY.

Description

For a subvariety X of an irreducible subscheme Y of \PP^{n_1}x...x\PP^{n_m} this command computes the algebraic multiplicity e_XY of X in Y. Let R be the coordinate ring of \PP^{n_1}x...x\PP^{n_m}, let O_{X,Y}=(R/I_Y)_{I_X} be the local ring obtained by localizing (R/I_Y) at the prime ideal I_X, and let len denote the length of a local ring. Let M be the unique maximal ideal of O_{X,Y}. The Hilbert-Samuel polynomial is the polynomial P_{HS}(t)=len(O_{X,Y}/M^t) for t large. In different words, this command computes the leading coefficient of the Hilbert-Samuel polynomial P_{HS}(t) associated to O_{X,Y}. Below we have an example of the multiplicity of the twisted cubic in a double twisted cubic.

i1 : R = ZZ/32749[x,y,z,w]

o1 = R

o1 : PolynomialRing

i2 : X = ideal(-z^2+y*w,-y*z+x*w,-y^2+x*z)

               2                        2
o2 = ideal (- z  + y*w, - y*z + x*w, - y  + x*z)

o2 : Ideal of R

i3 : Y = ideal(-z^3+2*y*z*w-x*w^2,-y^2+x*z)

               3               2     2
o3 = ideal (- z  + 2y*z*w - x*w , - y  + x*z)

o3 : Ideal of R

i4 : multiplicity(X,Y)

o4 = 2

o4 : QQ

For the programmer

The object multiplicity is a method function with options.