# getIdealFromNoetherianOperators -- Computes a primary ideal corresponding to a list of Noetherian operators and a prime ideal

## Synopsis

• Usage:
Q = getIdealFromNoetherianOperators(L, P)
• Inputs:
• Outputs:
• Q, an ideal, the primary ideal corresponding to L and P

## Description

This method contains an implementation of Algorithm 3.9 in Primary ideals and their differential equations. Applying this algorithm can be seen as making the backward process of computing a set of Noetherian operators.

Let $R$ be a polynomial ring $R = K[x_1,\ldots,x_n]$ over a field $K$ of characteristic zero. Consider the Weyl algebra $D = R<dx_1,\ldots,dx_n>$ and a list of differential operators $L = \{L_1,\ldots,L_m\} \,\subset\, D$. Denote by $\mathcal{E} \,\subset\, D$ the $R$-bisubmodule of $D$ that is generated by $L_1,\ldots,L_m$. For a given prime ideal $P \,\subset\, R$, this method computes the $P$-primary ideal given as $$Q = \{ f \,\in\, R\, \mid\, \delta\, \bullet\, f\, \in P \, \forall \delta \in \mathcal{E} \}.$$

This method can be seen as the reverse operation of computing a set of Noetherian operators for a primary ideal.

Next, we provide several examples to show the interplay between computing a set of Noetherian operators and then getting the back the original ideal.

The first example shows an ideal that can be described with two different sets of Noetherian operators (this example appeared in Example 7.8 of Primary ideals and their differential equations).

 i1 : R = QQ[x_1,x_2,x_3,x_4] o1 = R o1 : PolynomialRing i2 : MM = matrix {{x_3,x_1,x_2},{x_1,x_2,x_4}} o2 = | x_3 x_1 x_2 | | x_1 x_2 x_4 | 2 3 o2 : Matrix R <--- R i3 : P = minors(2,MM) 2 2 o3 = ideal (- x + x x , - x x + x x , - x + x x ) 1 2 3 1 2 3 4 2 1 4 o3 : Ideal of R i4 : M = ideal{x_1^2,x_2^2,x_3^2,x_4^2} 2 2 2 2 o4 = ideal (x , x , x , x ) 1 2 3 4 o4 : Ideal of R i5 : Q = joinIdeals(P,M); o5 : Ideal of R i6 : L1 = noetherianOperators(Q) -- A set of Noetherian operators 2 2 3 2 2 o6 = {1, dx_2, dx_1, dx_2 , dx_1*dx_2, dx_1 , dx_2 , dx_1*dx_2 , dx_1 dx_2, ------------------------------------------------------------------------ 3 3 2 2 3 dx_1 , 72x x dx_1 dx_2 + 180x x dx_1 dx_2 + 72x x dx_1*dx_2 } 1 3 2 3 3 4 o6 : List i7 : Q1 = getIdealFromNoetherianOperators(L1, P); o7 : Ideal of R i8 : Q == Q1 o8 = true i9 : L2 = noetherianOperators(M) -- Another set of Noetherian operators o9 = {1, dx_4, dx_3, dx_2, dx_1, dx_3*dx_4, dx_2*dx_4, dx_1*dx_4, dx_2*dx_3, ------------------------------------------------------------------------ dx_1*dx_3, dx_1*dx_2, dx_2*dx_3*dx_4, dx_1*dx_3*dx_4, dx_1*dx_2*dx_4, ------------------------------------------------------------------------ dx_1*dx_2*dx_3, dx_1*dx_2*dx_3*dx_4} o9 : List i10 : Q2 = getIdealFromNoetherianOperators(L2, P); o10 : Ideal of R i11 : Q == Q2 o11 = true

The following example was given as the running example in the Introduction of Primary ideals and their differential equations.

 i12 : Q = ideal(3*x_1^2*x_2^2-x_2^3*x_3-x_1^3*x_4-3*x_1*x_2*x_3*x_4+2*x_3^2*x_4^2,3*x_1^3*x_2*x_4-3*x_1*x_2^2*x_3*x_4-3*x_1^2*x_3*x_4^2+3*x_2*x_3^2*x_4^2+2*x_2^3-2*x_3*x_4^2,3*x_2^4*x_3-6*x_1*x_2^2*x_3*x_4+3*x_1^2*x_3*x_4^2+x_2^3-x_3*x_4^2,4*x_1*x_2^3*x_3+x_1^4*x_4-6*x_1^2*x_2*x_3*x_4-3*x_2^2*x_3^2*x_4+4*x_1*x_3^2*x_4^2,x_2^5-x_1*x_2^3*x_4-x_2^2*x_3*x_4^2+x_1*x_3*x_4^3,x_1*x_2^4-x_2^3*x_3*x_4-x_1*x_2*x_3*x_4^2+x_3^2*x_4^3,x_1^4*x_2-x_2^3*x_3^2-2*x_1^3*x_3*x_4+2*x_1*x_2*x_3^2*x_4,x_1^5-4*x_1^3*x_2*x_3+3*x_1*x_2^2*x_3^2+2*x_1^2*x_3^2*x_4-2*x_2*x_3^3*x_4,3*x_1^4*x_3*x_4-6*x_1^2*x_2*x_3^2*x_4+3*x_2^2*x_3^3*x_4+2*x_1^3*x_2+6*x_1*x_2^2*x_3-6*x_1^2*x_3*x_4-2*x_2*x_3^2*x_4,4*x_2^3*x_3^3+4*x_1^3*x_3^2*x_4-12*x_1*x_2*x_3^3*x_4+4*x_3^4*x_4^2-x_1^4+6*x_1^2*x_2*x_3+3*x_2^2*x_3^2-8*x_1*x_3^2*x_4) 2 2 3 3 2 2 3 2 o12 = ideal (3x x - x x - x x - 3x x x x + 2x x , 3x x x - 3x x x x - 1 2 2 3 1 4 1 2 3 4 3 4 1 2 4 1 2 3 4 ----------------------------------------------------------------------- 2 2 2 2 3 2 4 2 2 2 3 3x x x + 3x x x + 2x - 2x x , 3x x - 6x x x x + 3x x x + x - 1 3 4 2 3 4 2 3 4 2 3 1 2 3 4 1 3 4 2 ----------------------------------------------------------------------- 2 3 4 2 2 2 2 2 5 3 x x , 4x x x + x x - 6x x x x - 3x x x + 4x x x , x - x x x - 3 4 1 2 3 1 4 1 2 3 4 2 3 4 1 3 4 2 1 2 4 ----------------------------------------------------------------------- 2 2 3 4 3 2 2 3 4 3 2 3 x x x + x x x , x x - x x x - x x x x + x x , x x - x x - 2x x x 2 3 4 1 3 4 1 2 2 3 4 1 2 3 4 3 4 1 2 2 3 1 3 4 ----------------------------------------------------------------------- 2 5 3 2 2 2 2 3 4 + 2x x x x , x - 4x x x + 3x x x + 2x x x - 2x x x , 3x x x - 1 2 3 4 1 1 2 3 1 2 3 1 3 4 2 3 4 1 3 4 ----------------------------------------------------------------------- 2 2 2 3 3 2 2 2 3 3 6x x x x + 3x x x + 2x x + 6x x x - 6x x x - 2x x x , 4x x + 1 2 3 4 2 3 4 1 2 1 2 3 1 3 4 2 3 4 2 3 ----------------------------------------------------------------------- 3 2 3 4 2 4 2 2 2 2 4x x x - 12x x x x + 4x x - x + 6x x x + 3x x - 8x x x ) 1 3 4 1 2 3 4 3 4 1 1 2 3 2 3 1 3 4 o12 : Ideal of R i13 : L = noetherianOperators(Q) 2 o13 = {1, dx_1, dx_1 - 2x dx_2} 2 o13 : List i14 : Q' = getIdealFromNoetherianOperators(L, P); o14 : Ideal of R i15 : Q == Q' o15 = true

The next example was given by Palamodov to show that there exists primary ideals that cannot be described by using differential operators with constant coefficients.

 i16 : R = QQ[x_1, x_2, x_3] o16 = R o16 : PolynomialRing i17 : Q = ideal(x_1^2, x_2^2, x_1-x_2*x_3) 2 2 o17 = ideal (x , x , - x x + x ) 1 2 2 3 1 o17 : Ideal of R i18 : L = noetherianOperators(Q) o18 = {1, x dx_1 + dx_2} 3 o18 : List i19 : Q' = getIdealFromNoetherianOperators(L,radical Q) 2 2 o19 = ideal (x x - x , x , x x , x ) 2 3 1 2 1 2 1 o19 : Ideal of R i20 : Q == Q' o20 = true

For the last example we consider an ideal defined by using the join construction.

 i21 : R = QQ[x_1..x_9] o21 = R o21 : PolynomialRing i22 : MM = genericMatrix(R, 3, 3) o22 = | x_1 x_4 x_7 | | x_2 x_5 x_8 | | x_3 x_6 x_9 | 3 3 o22 : Matrix R <--- R i23 : P = minors(2, MM) o23 = ideal (- x x + x x , - x x + x x , - x x + x x , - x x + x x , - 2 4 1 5 3 4 1 6 3 5 2 6 2 7 1 8 ----------------------------------------------------------------------- x x + x x , - x x + x x , - x x + x x , - x x + x x , - x x + 3 7 1 9 3 8 2 9 5 7 4 8 6 7 4 9 6 8 ----------------------------------------------------------------------- x x ) 5 9 o23 : Ideal of R i24 : M = ideal(x_1^2, x_5^2, x_9^2, x_2, x_3, x_4, x_6, x_7, x_8) 2 2 2 o24 = ideal (x , x , x , x , x , x , x , x , x ) 1 5 9 2 3 4 6 7 8 o24 : Ideal of R i25 : Q = joinIdeals(P, M) 2 2 2 2 o25 = ideal (x x x - x x x , x x - 2x x x x + x x , x x x x - 2x x x x + 2 6 7 3 4 8 3 8 2 3 8 9 2 9 3 6 7 8 3 4 8 9 ----------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2 2 2 x x x , x x - 2x x x x + x x , x x - 2x x x x + x x , x x - 2 4 9 6 7 4 6 7 9 4 9 5 7 4 5 7 8 4 8 2 7 ----------------------------------------------------------------------- 2 2 2 2x x x x + x x , x x x - 2x x x x + x x x x , x x x x - 2x x x x + 1 2 7 8 1 8 3 5 7 3 4 5 8 2 4 6 8 2 3 4 7 1 3 4 8 ----------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2 2 2 x x x , x x - 2x x x x + x x , x x - 2x x x x + x x , x x x - 1 6 8 3 5 2 3 5 6 2 6 3 4 1 3 4 6 1 6 6 7 8 ----------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2 2x x x x - x x x + 2x x x x , x x x - 2x x x x - x x x + 4 6 8 9 5 7 9 4 5 8 9 3 6 8 2 6 8 9 3 5 9 ----------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2 2 2x x x x , 2x x x x - x x x - 2x x x x + x x x , x x x - 2x x x x 2 5 6 9 3 5 6 8 2 6 8 3 5 8 9 2 5 9 3 7 8 2 3 7 9 ----------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2 2 2 + 2x x x x - x x x , 2x x x x - x x x - 2x x x x + x x x , x x x - 1 2 7 9 1 8 9 5 6 7 8 4 6 8 5 6 7 9 4 5 9 3 6 7 ----------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2x x x x + 2x x x x - x x x , x x x - 2x x x x + 2x x x x - 3 4 7 9 1 3 4 9 1 6 9 3 4 7 1 3 6 7 1 6 7 9 ----------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2 2 x x x , x x x - 2x x x x + 2x x x x - x x x , 2x x x x - 2x x x x 1 4 9 2 3 7 1 3 7 8 1 3 8 9 1 2 9 2 4 5 7 1 2 5 7 ----------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2 2 2 - x x x + x x x , x x x - x x x - 2x x x x + 2x x x x , 2x x x x - 2 4 8 1 5 8 2 4 7 1 5 7 1 2 4 8 1 4 5 8 2 3 4 5 ----------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2 2 2 2x x x x - x x x + x x x , x x x - x x x - 2x x x x + 2x x x x , 1 3 4 5 2 4 6 1 5 6 2 3 4 1 3 5 1 2 4 6 1 2 5 6 ----------------------------------------------------------------------- 3 3 2 2 2 2 3 3 2 3 2 2 x x - 3x x x x + 3x x x x - x x , x x x + 2x x x x x - 4x x x x x 6 8 5 6 8 9 5 6 8 9 5 9 4 6 8 5 6 7 8 9 4 5 6 8 9 ----------------------------------------------------------------------- 3 2 2 2 3 2 2 2 2 - 2x x x + 3x x x x , x x x x - 2x x x x x - x x x x + 2x x x x x , 5 7 9 4 5 8 9 3 4 6 8 2 4 6 8 9 2 5 7 9 2 4 5 8 9 ----------------------------------------------------------------------- 3 2 2 2 3 2 2 2 2 x x x + 2x x x x x - 4x x x x x - 2x x x + 3x x x x , 2x x x x x - 2 6 8 3 5 6 8 9 2 5 6 8 9 3 5 9 2 5 6 9 1 3 4 6 8 ----------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2 2 2 2 x x x - 2x x x x x + x x x , 2x x x x x + x x x + 2x x x x x - 1 6 8 2 3 4 8 9 2 4 9 3 4 5 7 8 1 6 8 2 3 4 8 9 ----------------------------------------------------------------------- 2 2 2 2 2 8x x x x x x - 4x x x x x x + 4x x x x x + 4x x x x x - x x x , 1 3 4 5 8 9 1 2 4 6 8 9 1 5 6 8 9 1 2 4 5 9 1 5 9 ----------------------------------------------------------------------- 3 2 2 2 2 2 3 3 x x x x - 4x x x x x + 2x x x x x + 2x x x x x - x x x x , x x - 3 4 7 8 1 3 4 8 9 1 3 6 8 9 1 2 3 4 9 1 2 6 9 3 7 ----------------------------------------------------------------------- 2 2 2 2 3 3 2 2 3 2 3x x x x + 3x x x x - x x , x x x x - x x x x + 2x x x x x - 1 3 7 9 1 3 7 9 1 9 1 5 6 7 2 3 4 8 1 2 4 6 8 ----------------------------------------------------------------------- 2 2 2 3 2 3 2 2 2 2x x x x x , 2x x x x x - 2x x x + x x x - 4x x x x x + 3x x x x , 1 4 5 6 8 1 2 4 5 7 1 5 7 2 4 8 1 2 4 5 8 1 4 5 8 ----------------------------------------------------------------------- 2 2 3 3 2 2 2 2 3 3 2x x x x x - 2x x x + x x x - 4x x x x x + 3x x x x , x x - 1 2 3 4 5 1 3 5 2 4 6 1 2 4 5 6 1 2 5 6 2 4 ----------------------------------------------------------------------- 2 2 2 2 3 3 2 2 3 2 2 2 3x x x x + 3x x x x - x x , x x x + 2x x x x x - 4x x x x x x - 1 2 4 5 1 2 4 5 1 5 1 6 8 2 3 4 8 9 1 2 4 6 8 9 ----------------------------------------------------------------------- 2 2 2 2 2 2 2 3 2 2 2x x x x x - x x x x + 4x x x x x x , x x x x - 2x x x x x - 1 2 5 7 9 2 4 8 9 1 2 4 5 8 9 1 3 6 8 1 2 6 8 9 ----------------------------------------------------------------------- 3 2 2 2 2 3 2 2 2 x x x x + 2x x x x x , x x x + 2x x x x x x - 4x x x x x x - 2 4 7 9 1 2 4 8 9 1 6 8 2 3 4 6 8 9 1 2 4 6 8 9 ----------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2 2 2 2x x x x x - x x x x + 4x x x x x x , x x x x x - x x x x - 1 3 4 5 9 2 4 6 9 1 2 4 5 6 9 1 2 4 6 8 1 5 6 8 ----------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2 2 2 2 2 2 x x x x x + x x x x x + x x x x - x x x x x , 2x x x x x - x x x x 2 4 6 8 9 1 5 6 8 9 2 4 5 9 1 2 4 5 9 2 3 4 6 8 1 5 6 8 ----------------------------------------------------------------------- 2 2 2 2 2 - 2x x x x x x - 5x x x x x + 4x x x x x x x + x x x x x + 1 3 4 5 8 9 2 4 6 8 9 1 2 4 5 6 8 9 1 5 6 8 9 ----------------------------------------------------------------------- 2 2 2 2 2 2 3 2 2 3 2 2 3 2 4x x x x - 4x x x x x + x x x , x x x x - 2x x x x x - x x x x + 2 4 5 9 1 2 4 5 9 1 5 9 1 6 7 8 1 4 6 8 9 2 3 4 9 ----------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2 2 2 2 2 2x x x x x , x x x x - x x x x - 4x x x x x x + 4x x x x x x x + 1 2 4 6 9 2 4 6 8 1 5 6 8 2 4 5 6 8 9 1 2 4 5 6 8 9 ----------------------------------------------------------------------- 2 2 2 2 3 2 2 4 2 3x x x x - 4x x x x x + x x x ) 2 4 5 9 1 2 4 5 9 1 5 9 o25 : Ideal of R i26 : L = noetherianOperators(Q) 3 3 o26 = {1, dx_5, x dx_4 + x dx_7, - x dx_4 + x dx_8, (x x x + x x x + 5 8 4 8 4 5 9 4 5 9 ----------------------------------------------------------------------- 2 3 2 3 2 2 2 4 x x x x )dx_4*dx_7 + (x x x + x x x x + x x x )dx_7 + (x x x + x x 4 5 8 9 4 8 9 4 5 8 9 4 8 9 4 5 9 5 9 ----------------------------------------------------------------------- 2 2 2 3 3 2 + x x x )dx_4*dx_8 + (x x x x + x x x + x x x )dx_7*dx_8 - x x x dx_4 5 8 9 4 5 8 9 5 8 9 5 8 9 5 8 9 ----------------------------------------------------------------------- 3 2 + x x dx_7 - x x x dx_8, x dx_4*dx_5 + x dx_5*dx_7, - x dx_4*dx_5 + 5 9 4 5 9 5 8 4 ----------------------------------------------------------------------- 3 3 2 3 x dx_5*dx_8, (x x x + x x x + x x x x )dx_4*dx_5*dx_7 + (x x x + 8 4 5 9 4 5 9 4 5 8 9 4 8 9 ----------------------------------------------------------------------- 2 3 2 2 2 4 2 2 x x x x + x x x )dx_5*dx_7 + (x x x + x x + x x x )dx_4*dx_5*dx_8 + 4 5 8 9 4 8 9 4 5 9 5 9 5 8 9 ----------------------------------------------------------------------- 2 3 3 2 (x x x x + x x x + x x x )dx_5*dx_7*dx_8 - x x x dx_4*dx_5 + 4 5 8 9 5 8 9 5 8 9 5 8 9 ----------------------------------------------------------------------- 3 2 x x dx_5*dx_7 - x x x dx_5*dx_8} 5 9 4 5 9 o26 : List i27 : Q' = getIdealFromNoetherianOperators(L, radical Q) 2 2 2 2 o27 = ideal (x x x - x x x , x x - 2x x x x + x x , x x x x - 2x x x x + 2 6 7 3 4 8 3 8 2 3 8 9 2 9 3 6 7 8 3 4 8 9 ----------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2 2 2 x x x , x x - 2x x x x + x x , x x - 2x x x x + x x , x x - 2 4 9 6 7 4 6 7 9 4 9 5 7 4 5 7 8 4 8 2 7 ----------------------------------------------------------------------- 2 2 2 2x x x x + x x , x x x - 2x x x x + x x x x , x x x x - 2x x x x + 1 2 7 8 1 8 3 5 7 3 4 5 8 2 4 6 8 2 3 4 7 1 3 4 8 ----------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2 2 2 x x x , x x - 2x x x x + x x , x x - 2x x x x + x x , x x x - 1 6 8 3 5 2 3 5 6 2 6 3 4 1 3 4 6 1 6 6 7 8 ----------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2 2x x x x - x x x + 2x x x x , x x x - 2x x x x - x x x + 4 6 8 9 5 7 9 4 5 8 9 3 6 8 2 6 8 9 3 5 9 ----------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2 2 2x x x x , 2x x x x - x x x - 2x x x x + x x x , x x x - 2x x x x 2 5 6 9 3 5 6 8 2 6 8 3 5 8 9 2 5 9 3 7 8 2 3 7 9 ----------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2 2 2 + 2x x x x - x x x , 2x x x x - x x x - 2x x x x + x x x , x x x - 1 2 7 9 1 8 9 5 6 7 8 4 6 8 5 6 7 9 4 5 9 3 6 7 ----------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2x x x x + 2x x x x - x x x , x x x - 2x x x x + 2x x x x - 3 4 7 9 1 3 4 9 1 6 9 3 4 7 1 3 6 7 1 6 7 9 ----------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2 2 x x x , x x x - 2x x x x + 2x x x x - x x x , 2x x x x - 2x x x x 1 4 9 2 3 7 1 3 7 8 1 3 8 9 1 2 9 2 4 5 7 1 2 5 7 ----------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2 2 2 - x x x + x x x , x x x - x x x - 2x x x x + 2x x x x , 2x x x x - 2 4 8 1 5 8 2 4 7 1 5 7 1 2 4 8 1 4 5 8 2 3 4 5 ----------------------------------------------------------------------- 2 2 2 2 2 2 2 2 2 2 2 2x x x x - x x x + x x x , x x x - x x x - 2x x x x + 2x x x x , 1 3 4 5 2 4 6 1 5 6 2 3 4 1 3 5 1 2 4 6 1 2 5 6 ----------------------------------------------------------------------- 3 3 2 2 2 2 3 3 2 2 2 2 2 x x - 3x x x x + 3x x x x - x x , 2x x x x x + x x x + 2x x x x x 6 8 5 6 8 9 5 6 8 9 5 9 3 4 5 7 8 1 6 8 2 3 4 8 9 ----------------------------------------------------------------------- 2 2 2 2 2 - 8x x x x x x - 4x x x x x x + 4x x x x x + 4x x x x x - x x x , 1 3 4 5 8 9 1 2 4 6 8 9 1 5 6 8 9 1 2 4 5 9 1 5 9 ----------------------------------------------------------------------- 3 3 2 2 2 2 3 3 3 3 2 2 2 2 x x - 3x x x x + 3x x x x - x x , x x - 3x x x x + 3x x x x - 3 7 1 3 7 9 1 3 7 9 1 9 2 4 1 2 4 5 1 2 4 5 ----------------------------------------------------------------------- 3 3 x x ) 1 5 o27 : Ideal of R i28 : Q == Q' o28 = true

## Ways to use getIdealFromNoetherianOperators :

• "getIdealFromNoetherianOperators(List,Ideal)"

## For the programmer

The object getIdealFromNoetherianOperators is .