# amult -- Computes the arithmetic multiplicity of a module

## Synopsis

• Usage:
m = amult(M)
m = amult(I)
• Inputs:
• M, , a submodule of a free module
• I, an ideal, an ideal
• Outputs:
• m, an integer, the arithmetic multiplicity of the module or the ideal

## Description

Given a submodule $M$ of a free module $F$, one computes the arithmetic multiplicity of $M$ as the sum, along the associated primes of $F/M$, of the length of the largest submodule of finite length of the quotient $M/F$ localized at the associated prime. The arithmetic multiplicity is a fundamental invariant from a differential point of view as it yields the minimal size of a differential primary decomposition. For more details the reader is referred to the paper Primary Decomposition with Differential Operators.

 i1 : R = QQ[x1,x2,x3,x4] o1 = R o1 : PolynomialRing i2 : U = image matrix{{x1*x2,x2*x3,x3*x4,x4*x1}, {x1^2,x2^2,x3^2,x4^2}} o2 = image | x1x2 x2x3 x3x4 x1x4 | | x1^2 x2^2 x3^2 x4^2 | 2 o2 : R-module, submodule of R i3 : amult U o3 = 22 i4 : I = ideal( x1^3*x3^2-x2^5, x2^2*x4^3-x3^5, x1^5*x4^2-x2^7, x1^2*x4^5-x3^7 ) 5 3 2 5 2 3 7 5 2 7 2 5 o4 = ideal (- x2 + x1 x3 , - x3 + x2 x4 , - x2 + x1 x4 , - x3 + x1 x4 ) o4 : Ideal of R i5 : amult I o5 = 207

## Ways to use amult :

• "amult(Ideal)"
• "amult(Module)"

## For the programmer

The object amult is .