# 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 ℙn1x...xℙnm; makeProductRing builds the graded coordinate ring of ℙn1x...xℙnm.
• IY, an ideal, a multi-homogeneous primary ideal defining a closed subscheme of ℙn1x...xℙnm; makeProductRing builds the graded coordinate ring of ℙn1x...xℙnm.
• Optional inputs:
• Verbose =>
• Outputs:
• eXY, an integer, the algebraic (Hilbert-Samuel) multiplicity eXY 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 ℙn1x...xℙnm this command computes the algebraic multiplicity eXY of X in Y. Let R be the coordinate ring of ℙn1x...xℙnm, let OX,Y=(R/IY)IX be the local ring obtained by localizing (R/IY) at the prime ideal IX, and let len denote the length of a local ring. Let M be the unique maximal ideal of OX,Y. The Hilbert-Samuel polynomial is the polynomial PHS(t)=len(OX,Y/Mt) for t large. In different words, this command computes the leading coefficient of the Hilbert-Samuel polynomial PHS(t) associated to OX,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

## Ways to use multiplicity :

• multiplicity(Ideal,Ideal)