noetherianOperators -- Noetherian operators

Description

Let $R$ be a polynomial ring $R = K[x_1,\ldots,x_n]$ over a field $K$ of characteristic zero. Consider the Weyl algebra $D = R<dx_1,\ldots,dx_n>$, a prime ideal $P \subset R$ and a $P$-primary ideal. When this method is applied we obtain a finite list of differential operators $L_1,\ldots,L_m \in D$ such that $$ Q = \{ f \,\in\, R\, \mid\, L_i\, \bullet\, f\, \in P, \ \forall 1 \le i \le m \}. $$ We say that $\{L_1,\ldots,L_m\}$ is a set of Noetherian operators for the primary ideal $Q$. In the output of the algorithm we always have that $m$ (the number of Noetherian operators) is equal to the multiplicity of $Q$ over the prime ideal $P$.

i1 : R=QQ[x_1,x_2,x_3,x_4]

o1 = R

o1 : PolynomialRing

i2 : Q = ideal(x_1^2,x_1*x_2,x_1*x_3,x_1*x_4-x_3^2+x_1,x_3^2*x_4-x_2^2,x_3^2*x_4-x_3^2-x_2*x_3+2*x_1)

             2                 2               2      2   2             2
o2 = ideal (x , x x , x x , - x  + x x  + x , x x  - x , x x  - x x  - x  +
             1   1 2   1 3     3    1 4    1   3 4    2   3 4    2 3    3  
     ------------------------------------------------------------------------
     2x )
       1

o2 : Ideal of R

i3 : isPrimary Q

o3 = true

i4 : noetherianOperators(Q, Strategy => "PunctualQuot")

o4 = {| 1 |, | dx_2 |, | dx_3 |, |
     ------------------------------------------------------------------------
     (x_4^2+x_4)dx_2^2+(2x_4^2+2)dx_2dx_3+(x_4+1)dx_3^2+2dx_1 |}

o4 : List

Ways to use noetherianOperators :

noetherianOperators(Ideal) -- Noetherian operators of a primary ideal
noetherianOperators(Ideal,Ideal) -- Noetherian operators of a primary component
noetherianOperators(Module) -- Noetherian operators of a primary submodule
noetherianOperators(Module,Ideal) -- Noetherian operators of a primary component

For the programmer

The object noetherianOperators is a method function with options.