# 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 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 => "PunctualHilbert") 2 2 2 2 o4 = {1, dx_3, dx_2, (2x + 2x )dx_2 + (4x + 4)dx_2*dx_3 + (2x + 2)dx_3 + 4 4 4 4 ------------------------------------------------------------------------ 4*dx_1} o4 : List

## For the programmer

The object noetherianOperators is .