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 |
The object amult is a method function.