amult -- Computes the arithmetic multiplicity of a module

Synopsis

Usage:

m = amult(M)

m = amult(I)
Inputs:
- M, a module, 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 a method function.