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# 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.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NoetherianOperators.m2:2836:0.