---
abstract: 'An ideal in a polynomial ring encodes a system of linear partial differential equations with constant coefficients. Primary decomposition organizes the solutions to the PDE. This paper develops a novel structure theory for primary ideals in this context. We characterize primary ideals in terms of PDE, punctual Hilbert schemes, and the join construction, and we present an explicit algorithm for computing Noetherian operators.'
author:
- 'Yairon Cid-Ruiz, Roser Homs and Bernd Sturmfels'
title: Primary Ideals and their Differential Equations
---

Introduction {#sec1}
============

In his 1938 article [@GROBNER_MATH_ANN] on the foundations of algebraic geometry, Gröbner introduced differential operators to characterize membership in a polynomial ideal. He derived such characterizations for ideals that are prime or primary to a rational maximal ideal [@GROBNER_BOOK_AG_2 pages 174-178]. In a 1952 lecture [@GROBNER_LIEGE §1] he suggested that the same program can be carried out for any primary ideal. Gröbner was particularly interested in algorithmic solutions to this problem.

Substantial contributions in this subject area were made by analysts. In the 1960s, Ehrenpreis [@EHRENPREIS] stated his *Fundamental Principle* on solutions to linear partial differential equations (PDE) with complex constant coefficients. A main step was the characterization of primary ideals by differential operators. But, he incorrectly claimed that operators with constant coefficients suffice. Using Example (\[exam:Palamodov\]) below, Palamodov [@PALAMODOV] pointed out the error, and he gave a correct proof by introducing the representation by [*Noetherian operators*]{}. Details on the Ehrenpreis-Palamodov Fundamental Principle can also be found in [@BJORK; @HORMANDER].

The ball returned to algebra in 1978 when Brumfiel published the little-known paper [@BRUMFIEL_DIFF_PRIM]. In 1999, Oberst [@OBERST_NOETH_OPS] extended Palamodov’s Noetherian operators to polynomial rings over arbitrary fields. In 2007, Damiano, Sabadini and Struppa [@DAMIANO] gave a computational approach. A general theory for Noetherian commutative rings was developed recently in [@NOETH_OPS]. Building on this, the present article develops a theory of primary ideals as envisioned by Gröbner.

We now introduce a running example that serves to illustrate our title and results. The following prime ideal of codimension $c=2$ in $n=4$ variables is familiar to many algebraists: $$\label{eq:twistedcubic1}
 P \quad = \quad \langle\,
x_1^2-x_2 x_3,\, x_1 x_2 - x_3 x_4, x_2^2 - x_1 x_4 \,\rangle \quad \subset
\quad {\mathbb{C} }[x_1,x_2,x_3,x_4].$$ This ideal defines the (affine cone over the) [*twisted cubic curve*]{} $\,V(P) = \bigl\{ \,(s^2t, s t^2,s^3, t^3) \,:\, s,t \in {\mathbb{C} }\,\bigr\}$; see [@CUBIC_LITTLE]. We identify the polynomials in (\[eq:twistedcubic1\]) with PDE with constant coefficients by setting $x_i = \partial_{z_i}$. Solving these PDE means describing all functions $\psi(z_1,z_2,z_3,z_4)$ with $$\label{eq:twistedcubic2}
 \frac{\partial^2 \psi}{\partial z_1^2} = \frac{\partial^2 \psi}{\partial z_2 \partial z_3}    \qquad {\rm and} \qquad
 \frac{\partial^2 \psi}{\partial z_1 \partial z_2} = \frac{\partial^2 \psi}{\partial z_3 \partial z_4}   \qquad {\rm and} \qquad
 \frac{\partial^2 \psi}{\partial z_2^2} = \frac{\partial^2 \psi}{\partial z_1 \partial z_4}.$$ Results in analysis ensure that every solution comes from a measure $\mu$ on the $(s,t)$-plane: $$\label{eq:twistedcubic4}
\psi(z_1,z_2,z_3,z_4) \,\,\,\, = \,\,\,
\int {\rm exp} \bigl(   z_1 s^2 t \,+\, z_2 s t^2 \,+\,z_3 s^3 \,+\, z_4 t^3
\bigr) \,\mu(s,t) \,{\rm d}s \,{\rm d}t .$$ For instance, if $\mu$ is the Dirac measure at the point $(2,3)$ then $\psi = {\rm exp}(  12 z_1 + 18 z_2 + 8 z_3 + 27 z_4)$. Thus, the functions $\psi$ are simply an analytic encoding of the affine surface $V(P) \subset {\mathbb{C} }^4$.

The situation becomes interesting when we consider a non-reduced scheme structure on our surface. Algebraically, this means replacing the prime $P$ by a $P$-primary ideal. We use differential operators to give compact representations of $P$-primary ideals $Q$. For instance, $$\label{eq:twistedcubic6}
\begin{matrix}
 Q \,\,=\, \, \bigl\{ \,f \in {\mathbb{C} }[x_1,x_2,x_3,x_4]\,:\,
A_i \bullet f  \in P \,\,\,{\rm for} \,\,\, i=1,2,3\, \bigr\}, \smallskip \\ {\rm where} \quad
\,A_1 \,=\, 1\,,\;\, A_2\,=\, \partial_{x_1}
\,\,\,{\rm and} \,\,\,A_3 \,=\, \partial_{x_1}^2 \,-\, 2 \,x_2\,\partial_{x_2} . \quad 
\end{matrix}$$ Here $\bullet$ means applying a differential operator to a function. Note that a prime ideal is always represented by just one Noetherian operator $A_1=1$. We can encode (\[eq:twistedcubic6\]) by the ideal $$\label{eq:magic}
\!\! \bigl\langle 
u_1^2 - u_2 u_3,  u_1 u_2 -u_3 u_4, 
u_2^2-u_1 u_4, 
\, x_1-u_1-y_1, x_2-u_2-y_2, x_3-u_3, x_4-u_4, \,
\underline{ y_1^3,     \, y_2 + u_2 \,y_1^2}
      \bigr\rangle.$$ The minimal generators of $Q$ are obtained from (\[eq:magic\]) by eliminating $\{u_1,u_2,u_3,u_4,y_1,y_2\}$: $$\begin{small} \begin{matrix} Q \,\, = & \!\!\! \bigl\langle\,
3 x_1^2 x_2^2-x_2^3 x_3-x_1^3 x_4-3 x_1 x_2 x_3 x_4+2 x_3^2 x_4^2\,,\,\, 
3 x_1^3 x_2 x_4-3 x_1 x_2^2 x_3 x_4-3 x_1^2 x_3 x_4^2+3 x_2 x_3^2 x_4^2\\ & +2 x_2^3 -2 x_3 x_4^2\,,\,\,
3 x_2^4 x_3-6 x_1 x_2^2 x_3 x_4+3 x_1^2 x_3 x_4^2+x_2^3-x_3 x_4^2\,,\,\, 
4 x_1 x_2^3 x_3+x_1^4 x_4-6 x_1^2 x_2 x_3 x_4\\ & -3 x_2^2 x_3^2 x_4+4 x_1 x_3^2 x_4^2\,,\,\,
x_2^5-x_1 x_2^3 x_4-x_2^2 x_3 x_4^2+x_1 x_3 x_4^3\,,\,\,  
x_1 x_2^4-x_2^3 x_3 x_4-x_1 x_2 x_3 x_4^2+x_3^2 x_4^3\,,\\ & 
x_1^4 x_2-x_2^3 x_3^2-2 x_1^3 x_3 x_4+2 x_1 x_2 x_3^2 x_4\,,\,\, 
x_1^5-4 x_1^3 x_2 x_3+3 x_1 x_2^2 x_3^2+2 x_1^2 x_3^2 x_4-2 x_2 x_3^3 x_4\,,\\ &
3 x_1^4 x_4^2-6 x_1^2 x_2 x_3 x_4^2+3 x_2^2 x_3^2 x_4^2+4 x_2^4-4 x_2 x_3 x_4^2\,,\,\, 
x_2^3 x_3^2 x_4+x_1^3 x_3 x_4^2-3 x_1 x_2 x_3^2 x_4^2+x_3^3 x_4^3 \\ & +x_1 x_2^3-x_1 x_3 x_4^2\,, 
3 x_1^4 x_3 x_4-6 x_1^2 x_2 x_3^2 x_4+3 x_2^2 x_3^3 x_4+2 x_1^3 x_2
+6 x_1 x_2^2 x_3-6 x_1^2 x_3 x_4-2 x_2 x_3^2 x_4\,,\\ &
4 x_2^3 x_3^3+4 x_1^3 x_3^2 x_4-12 x_1 x_2 x_3^3 x_4+4 x_3^4 x_4^2 
-x_1^4+6 x_1^2 x_2 x_3+3 x_2^2 x_3^2-8 x_1 x_3^2 x_4\,
\bigr\rangle.
\end{matrix} \end{small}$$ As in (\[eq:twistedcubic1\]) and (\[eq:twistedcubic2\]), we can view $Q$ as a system of PDE by setting $x_i = \partial_{z_i}$. Its solutions are $$\label{eq:twistedcubic5}
\begin{matrix}
\!\! \psi(z_1,z_2,z_3,z_4) \, = \,\,
\sum_{i=1}^3 \int \! B_i(z_1,z_2,s,t) \cdot
{\rm exp} \bigl(  z_1 s^2 t + z_2 s t^2 +z_3 s^3 +  z_4 t^3 \bigr)  \,\mu_i(s,t) \,{\rm d}s \,{\rm d}t ,
\smallskip \\ \qquad {\rm where}  \quad
 B_1\,=\, 1\,,\;\, B_2 = z_1 \,\,\, {\rm and} \,\,\, B_3 \,=\, z_1^2  -  2 st^2 z_2 ,
\end{matrix}$$ for suitable measures $\mu_1,\mu_2,\mu_3$ on the $(s,t)$-plane ${\mathbb{C} }^2$. Note that $Q$ has multiplicity $3$ over $P$.

The title of this paper refers to two ways of associating differential equations to a primary ideal in a polynomial ring. First, we use PDE with polynomial coefficients, namely Noetherian operators $A_i$ as in (\[eq:twistedcubic6\]), to give a compact encoding of $Q$. Second, we can interpret $Q$ itself as a system of PDE with constant coefficients, with solutions represented by [*Noetherian multipliers*]{} $B_i$ as in (\[eq:twistedcubic5\]). The dual roles played by the $A_i$ and $B_i$ is one of our main themes.

This paper is organized as follows. In Section \[sec2\] we present characterizations of primary ideals in terms of punctual Hilbert schemes and Weyl-Noether modules. The former offers a parametrization of all $P$-primary ideals of a given multiplicity, and the latter establishes the links to differential equations. In Section \[sec3\] we turn to the Ehrenpreis-Palamodov Fundamental Principle. We present a self-contained proof of the algebraic part, and we introduce algorithms for computing Noetherian operators. In Sections \[sec4\] and \[sec6\] we prove the results stated in Section \[sec2\]. Section \[sec5\] reviews differential operators in commutative algebra and supplies tools for our proofs. In Section \[sec7\] we study the join construction for primary ideals, which offers a new perspective on ideals that are similar to symbolic powers. Finally, in Section \[sec8\] we establish a connection to numerical algebraic geometry. We propose a definition of [*numerical primary decomposition*]{} that puts a focus on the representation of primary ideals.

Characterizing Primary Ideals {#sec2}
=============================

Irreducible varieties and their prime ideals are the basic building blocks in algebraic geometry. Solving systems of polynomial equations means extracting the associated primes from the system, and to subsequently study their irreducible varieties. However, if the given ideal is not radical then we seek the primary decomposition and not just the associated primes. We wish to gain a precise understanding of the primary ideals that make up the given scheme.

We furnish a representation theorem for primary ideals in a polynomial ring, extending the familiar case of zero-dimensional ideals (Macaulay’s inverse system [@Grobner37]). This combines a characterization via differential operators with a parametrization from a Hilbert scheme. Fix a field ${\mathbb{K}}$ of characteristic zero and a prime ideal $P$ of codimension $c$ in the polynomial ring $R = {\mathbb{K}}[x_1,\ldots,x_n]$. We write ${\mathbb{F}}$ for the field of fractions of the integral domain $R/P$.

\[thm:main\] The following four sets of objects are in a natural bijective correspondence:

(a) $P$-primary ideals $Q$ in $R$ of multiplicity $m$ over $P$,

(b) points in the punctual Hilbert scheme $\,{\rm Hilb}^m({\mathbb{F}}[[y_1,\ldots,y_c]])$,

(c) $m$-dimensional ${\mathbb{F}}$-subspaces of $\,{\mathbb{F}}[z_1,\ldots,z_c]$ that are closed under differentiation,

(d) $m$-dimensional ${\mathbb{F}}$-subspaces of the Weyl-Noether module $ {\mathbb{F}}\,\otimes_R  \,D_{n,c}$ that are $R$-bi-modules.

Moreover, any basis of the ${\mathbb{F}}$-subspace in part (d) can be lifted to Noetherian operators $A_1,\ldots,A_m$ in the relative Weyl algebra $D_{n,c}$ that represent the ideal $Q$ in part (a) as in (\[eq:fromAtoQ\]).

The purpose of this section is to define and explain all the concepts in Theorem \[thm:main\]. Our aim is to state the promised bijections as explicitly as possible. The proof of Theorem \[thm:main\] will be divided into smaller pieces and given in Sections \[sec4\] and \[sec6\]. The encoding of $Q$ by Noetherian operators $A_i$ will be explained in Section \[sec3\]. We already saw an example in (\[eq:twistedcubic6\]). The Weyl-Noether module in part (d) is our stage for the PDE that portray primary ideals.

We begin by returning to Gröbner, whose 1937 article [@Grobner37] interpreted Macaulay’s inverse system as solutions to linear PDE. He considered the special case when $P = \langle x_1,\ldots,x_n \rangle $ is the maximal irrelevant ideal, so we have $c=n$ and ${\mathbb{F}}= {\mathbb{K}}$. The geometric intuition invoked in [@GROBNER_LIEGE §1] is captured by the punctual Hilbert scheme $\,{\rm Hilb}^m({\mathbb{K}}[[y_1,\ldots,y_n]])$, whose points are precisely the $P$-primary ideals of colength $m$. This zero-dimensional case is familiar to most commutative algebraists, especially the readers of [@MOURRAIN_DUALITY]. Here, parts (c) and (d) of Theorem \[thm:main\] refer to the $m$-dimensional ${\mathbb{K}}$-vector space of polynomial solutions to the PDE.

The general case of higher-dimensional primary ideals $Q$ was of great interest to Gröbner. In his 1952 Liège lecture [@GROBNER_LIEGE], he points to Severi [@Severi], and he writes: [*En ce sense la variété algébraique correspondante à un idéal primaire $Q$ pour l’idéal premier $P$ consiste en les points ordinaires de la variété $\,V(P)$ et en certain nombre $m$ des points infinitesiment voisins, c’est-à-dire dans $m$ conditions différentielles ajoutées à chaque point de la variété $\,V(P)$. Le nombre $m$ de ces conditions différentielles est égal à la longueur de l’idéal primaire $Q$.*]{} But Gröbner was never able to complete the program himself, in spite of the optimism he still expressed in his 1970 textbook [@GROBNER_BOOK_AG_2]. After the detailed treatment of Macaulay’s inverse systems for zero-dimensional ideals, he proclaims: [*Es dürfte auch nicht schwer sein den oben angegebenen Formalismus auf mehrdimensionale Primärideale auszudehnen*]{} [@GROBNER_BOOK_AG_2 page 178].

The issue was finally resolved by the theory of Ehrenpreis-Palamodov [@EHRENPREIS; @PALAMODOV], presented in Section \[sec3\], and the subsequent developments [@BRUMFIEL_DIFF_PRIM; @NOETH_OPS; @DAMIANO; @OBERST_NOETH_OPS] we discussed in the Introduction.

Theorem \[thm:main\] is our main contribution. We regard this as a definitive result on primary ideals in $R$. It captures the geometric spirit of Gröbner and Severi, as it relates their “infinitely near points” directly to current advances in numerical algebraic geometry (Section \[sec8\]).

Two essential ingredients in Theorem \[thm:main\] are the function field ${\mathbb{F}}$ and the Weyl-Noether module $\, {\mathbb{F}}\otimes_R D_{n,c}$. We start our technical discussion with some insights into these objects. By [*Noether normalization*]{}, after a linear change of coordinates, the quotient ring $R/P$ is a finitely generated module over the polynomial subring ${\mathbb{K}}[x_{c+1},\ldots,x_n]$. This implies that ${\mathbb{F}}$ is algebraic over the field ${\mathbb{K}}(x_{c+1},\ldots,x_n)$, a purely transcendental extension of ${\mathbb{K}}$.

Clear notation is very important for this article. This is why multiple letters $x,y,z,u$ are used to denote variables and differential operators. Elements in ${\mathbb{F}}$ are represented as fractions of polynomials in ${\mathbb{K}}[u_1,\dots,u_n]$, where $u_i$ denotes the residue class of $x_i$ modulo $P$. Whenever the number $n$ of variables is clear from the context, we use the multi-index notation $\mathbf{u}^\alpha=u_1^{\alpha_1}\cdots u_n^{\alpha_n}$. Elements $a(\mathbf{u})/b(\mathbf{u})$ of the field ${\mathbb{F}}$ can be uniquely represented by taking $a(\mathbf{u})$ and $b(\mathbf{u})$ coprime and in normal form with respect to a Gröbner basis of $P$. Arithmetic in ${\mathbb{F}}$ is performed via this Gröbner basis. The $R$-module structure of ${\mathbb{F}}$ is given by $\,\mathbf{x}^\alpha\cdot a(\mathbf{u})/b(\mathbf{u})=\mathbf{u}^\alpha a(\mathbf{u})/b(\mathbf{u})$. Alternatively, from the perspective of numerical algebraic geometry, a better approach to arithmetic in ${\mathbb{F}}$ is to work with generic points, obtained by realizing $R/P$ as a subring of a suitable field of functions on $V(P)$. In our running example, that suitable field could be $\,{\mathbb{K}}(s/t,t^3)$. It contains $R/P$ as the subring ${\mathbb{K}}[s^2 t, st^2, s^3, t^3]$.

The [*relative Weyl algebra*]{} $D_{n,c} = {\mathbb{K}}\langle x_1,\ldots,x_n, \partial_{x_1},\dots,\partial_{x_c}\rangle$ is the ${\mathbb{K}}$-algebra on $n{+}c$ generators $x_1,\ldots,x_n,\partial_{x_1},\ldots,\partial_{x_c}$ that commute except for $\partial_{x_i}x_i=x_i\partial_{x_i}+1$. This is a subalgebra of the usual Weyl algebra, so $D_{n,c}$ is non-commutative. Its elements are linear differential operators with polynomial coefficients, where derivatives occur with respect to the first $c$ variables. The set $\left\lbrace x_1^{\alpha_1}\cdots x_n^{\alpha_n}{\partial_{x_1}}^{\!\!\!\!\beta_1}\cdots
 {\partial_{x_c}}^{\!\!\!\!\beta_c}:(\alpha,\beta)\in\mathbb{N}^n\times\mathbb{N}^c\right\rbrace$ is a ${\mathbb{K}}$-basis of $D_{n,c}$.

We define the [*Weyl-Noether module*]{} of the affine variety $V(P)$ to be the tensor product $$\label{eq:relativeweyl}
{\mathbb{F}}\,\otimes_R\,D_{n,c} \,\, = \,\, {\mathbb{F}}\,\otimes_R \,
R \langle \partial_{x_1},\ldots,\partial_{x_c}\rangle.$$ Since ${\mathbb{F}}$ is the field of fractions of the integral domain $R/P$, it is clearly an $R$-module. Note that the relative Weyl algebra $D_{n,c} = R\langle \partial_{x_1},\dots,\partial_{x_c}\rangle$ is non-commutative, and it has two distinct $R$-module structures: it is a left $R$-module and it is a right $R$-module. In the tensor product (\[eq:relativeweyl\]), for convenience of notation, we mean the left $R$-module structure on $D_{n,c}$. Later, in Remark \[rem\_isom\_restrict\_Weyl\_mod\], we shall give an intrinsic description of ${\mathbb{F}}\otimes_R D_{n,c}$ with differential operators.

By construction, the Weyl-Noether module (\[eq:relativeweyl\]) has both right and left $R$-module structures. The action by $R$ on the left is easy to write using the standard ${\mathbb{K}}$-basis above: $$\label{eq:leftaction}
\mathbf{x}^\alpha \cdot \biggl( \frac{a(\mathbf{u})}{b(\mathbf{u})} \,\otimes_R\, \partial_{\mathbf{x}}^\beta\ \biggr) \,\,\, = \,\,\,
\frac{\mathbf{u}^{\alpha} a(\mathbf{u})}{b(\mathbf{u})} \,\otimes_R\, \partial_{\mathbf{x}}^\beta.$$ For the action on the right we need the commutation identities in the Weyl algebra: $$\partial_{\mathbf{x}}^\beta \mathbf{x}^\alpha\,\,=\,\,\sum_{\gamma, \delta} \lambda_{\gamma,\delta}
\,  \mathbf{x}^\gamma \partial_\mathbf{x}^\delta.$$ Here $\lambda_{\gamma,\delta}$ are the positive integers derived in [@SatStu Problem 4]. With this, the right action is $$\label{eq:rightaction}
 \biggl( \frac{a(\mathbf{u})}{b(\mathbf{u})}  \,\otimes_R \, \partial_{\mathbf{x}}^\beta \biggr)
 \cdot \mathbf{x}^\alpha  \, \,\, = \,\,\, \frac{a(\mathbf{u})}{b(\mathbf{u})}
 \, \otimes_R \, \partial_{\mathbf{x}}^\beta \mathbf{x}^\alpha \,\,\,=\,\,\,
\sum_{\gamma,\delta} \lambda_{\gamma,\delta} \, \frac{\mathbf{u}^{\gamma}a(\mathbf{u})}{b(\mathbf{u})}  \, \otimes_R \, 
\partial_\mathbf{x}^\delta.$$ This means that the requirement to be an $R$-bi-module in Theorem \[thm:main\] (d) is very strong.

From the action (\[eq:leftaction\]) we deduce that $\,{\mathbb{F}}\,\otimes_R\,D_{n,c}\,$ is a left ${\mathbb{F}}$-vector space with basis $\,\left\lbrace 1 \otimes_R \partial_{\mathbf{x}}^\beta: \beta\in\mathbb{N}^c\right\rbrace$, so we could also write ${\mathbb{F}}\langle \partial_{x_1},\dots,\partial_{x_c}\rangle$ for (\[eq:relativeweyl\]). However, we prefer the previous notation because it highlights that there are two distinct structures. The Weyl-Noether module is a left ${\mathbb{F}}$-vector space via (\[eq:leftaction\]) and it is a right $R$-module via (\[eq:rightaction\]). It is not a right ${\mathbb{F}}$-vector space because the right $R$-action is not compatible with passing to $R/P$:

Fix the maximal ideal $P=\langle x_1, \ldots,x_n\rangle$ so that ${\mathbb{F}}={\mathbb{K}}$ and $c=n$. Since $\overline{x_j} = 0 \in R/P$, we have $\,x_j \cdot \left(1 \otimes_R \partial_{x_j}\right) = 0 \,$ and hence $\,\left(1 \otimes_R \partial_{x_j}\right) \cdot x_j = 1 \otimes_R 1$ holds in $\,{\mathbb{F}}\,\otimes_R\,D_{n,c}$. This shows that there is no right ${\mathbb{F}}$-action on the Weyl-Noether module $\,{\mathbb{F}}\,\otimes_R\,D_{n,c}$.

We now come to our parameter space in part (b), namely the punctual Hilbert scheme $$\label{eq:hilbertscheme}
 {\rm Hilb}^m \bigl( \,{\mathbb{F}}[[y_1,\ldots,y_c]] \,\bigr).$$ This is a quasiprojective scheme over the function field ${\mathbb{F}}$. Its classical points are ideals of colength $m$ in the local ring ${\mathbb{F}}[[y_1,\ldots,y_c]]$. By Cohen’s Structure Theorem, this ring is the completion of $R_P$, the localization of $R$ at the prime $P$. To connect parts (a) and (b), we recall that the multiplicity $m$ of a primary ideal $Q$ over its prime $P = \sqrt{Q}$ is the length of the artinian local ring $\,R_P/Q R_P$. In symbols, using the command [degree]{} in [Macaulay2]{} [@MAC2], $$m \,\,=\,\, {\rm length}\bigl( R_P/QR_P \bigr) \,\,=\,\, \frac{{\tt degree}(Q)}{{\tt degree}(P)}.$$

The punctual Hilbert scheme (\[eq:hilbertscheme\]) is familiar to algebraic geometers, but its structure is very complicated when $c \geq 3$. We refer to Iarrobino’s article [@IARROBINO_HILB] as a point of entry. While the punctual Hilbert scheme is trivial for $c=1$, Briançon [@Bri77] undertook a detailed study for $c=2$. He showed that $\,{\rm Hilb}^m \bigl( {\mathbb{F}}[[y_1,y_2]] \bigr)\,$ is smooth and irreducible of dimension $m-1$. A dense subset is given by the $(m-1)$-dimensional family of $\langle y_1,y_2 \rangle$-primary ideals of the form $$\label{eq:HSfamily}
 \qquad
 \bigl\langle  \,\, y_1^m\,,\, \,y_2 \,+\, a_1 y_1 + a_2 y_1^2 +  \cdots +  a_{m-1} y_1^{m-1} \,
 \bigr\rangle, \qquad
 {\rm where}\,\,a_1,a_2,\ldots, a_{m-1} \in {\mathbb{F}}.$$ For instance, for $m=3$, the Hilbert scheme (\[eq:hilbertscheme\]) is a surface over ${\mathbb{F}}$. Each of its points encodes a scheme structure of multiplicity $3$ on the variety $V(P)$. This is the generic point on $V(P)$ together with two “infinitely near points”, in the language of Gröbner and Severi.

To see that the family (\[eq:HSfamily\]) is a proper subset of $\,{\rm Hilb}^m \bigl( {\mathbb{F}}[[y_1,y_2]] \bigr)$, we consider the points $$\qquad \langle\,  y_1^3 \,, \,y_2 + \epsilon^{-1} y_1^2 \,\rangle\,\,  =\,\,
\langle \,y_1^2 + \epsilon y_2\, , \,y_1y_2\, ,\,y_2^2 \,\rangle \quad \in \,\,\,
{\rm Hilb}^3 \bigl( {\mathbb{F}}[[y_1,y_2]] \bigr).$$ For $\epsilon \in {\mathbb{F}}\backslash \{0\}$, this $\langle y_1,y_2 \rangle $-primary ideal is in the family (\[eq:HSfamily\]), but for $\epsilon = 0$ it is not.

In the zero-dimensional case, when $P = \langle x_1,\ldots,x_n \rangle$, the correspondences in Theorem \[thm:main\] are well-known since the 1930’s. Wolfgang Gröbner tells us: [*Die noch verbleibende Aufgabe, die Integrale eines Primärideals aus denjenigen für das zugehörige Primideal abzuleiten, wollen wir hier wenigstens für null-dimensionale Primärideale allgemein lösen*]{} [@Grobner39 page 272]. In our current understanding, the $P$-primary ideals are points in ${\rm Hilb}^m({\mathbb{K}}[[y_1,\dots,y_n]])$, subspaces closed by differentiation are Macaulay’s inverse systems, and these account for polynomial solutions to linear PDE with constant coefficients [@MOURRAIN_DUALITY; @STURMFELS_SOLVING].

The idea behind Theorem \[thm:main\] is to reduce the study of arbitrary primary ideals in $R = {\mathbb{K}}[x_1,\ldots,x_n]$ to a zero-dimensional setting over the function field ${\mathbb{F}}$. Recall that coordinates were chosen so that $R/P$ is finite over ${\mathbb{K}}[x_{c+1},\ldots,x_n]$. We define the inclusion map $$\label{eq_map_gamma}
\gamma:R \hookrightarrow {\mathbb{F}}[y_1,\dots,y_c]\, , \qquad
\begin{matrix}
x_i  &\mapsto &  y_i+u_i, & \!\!\!\!\! \mbox{ for }1\leq i\leq c,\\
x_j & \mapsto  & u_j,& \quad \mbox{ for }c+1\leq j\leq n,
\end{matrix}$$ where $u_i$ denotes the class of $x_i$ in ${\mathbb{F}}$, for $1\leq i\leq n$. With this, we can give an explicit description of the correspondence between the objects in parts (a) and (b) of Theorem \[thm:main\]: $$\label{eq:corr12}
\begin{array}{ccc}
\left\lbrace\begin{array}{c}
\mbox{$P$-primary ideals of $R$}\\
\mbox{with multiplicity $m$ over $P$}
\end{array}\right\rbrace
& \longleftrightarrow &
\left\lbrace\begin{array}{c}
\mbox{points in }{\rm Hilb}^m({\mathbb{F}}[[y_1,\ldots,y_c]])\\
\end{array}\right\rbrace\\
Q & \longrightarrow & I=\langle y_1,\dots,y_c\rangle^m+\gamma(
Q){\mathbb{F}}[y_1,\dots,y_c]\\
Q\,=\,\gamma^{-1}(I) & \longleftarrow & I.
\end{array}$$

\[Bij:1,2\] Fix $P $ and $Q$ as in the Introduction, with $n=4$, $m=3$, $c=2$, where $R/P$ is finite over ${\mathbb{C} }[x_3,x_4]$. The primary ideal $Q$ corresponds to a point in ${\rm Hilb}^3({\mathbb{F}}[[y_1,y_2]])$. See [@Bri77 Section IV.2] for a detailed description of points in the Hilbert scheme of degree 3 in two variables. The bijection in (\[eq:corr12\]) gives us the following point in the punctual Hilbert scheme: $$\label{eq:punctual2}
 I \,\,=\,\,\langle y_2^2,y_1y_2,y_1^2+u_2^{\,-1}y_2\rangle
 \,\,\,\subset \,\,{\mathbb{F}}[[y_1,y_2]].$$ Note that this ideal is also generated by $y_1^3$ and $y_2+u_2y_1^2$, as in (\[eq:magic\]).

The bijection between (b) and (c) is Macaulay’s duality between zero-dimensional ideals in a power series ring and finite-dimensional subspaces in a polynomial ring that are closed under differentiation. To interpret polynomials in $I$ as PDE, we replace $y_i$ by $\partial_{z_i}$. So, by slight abuse of notation, we shall write ${\mathbb{F}}[[y_1,\ldots,y_c]]$ and ${\mathbb{F}}[[\partial_{z_1}, \ldots, \partial_{z_c}]]$ interchangeably. With this, the [*inverse system*]{} of a zero-dimensional ideal $I$ in the local ring ${\mathbb{F}}[[\partial_{z_1},\dots,\partial_{z_c}]]$, denoted by $I^\perp$, is the ${\mathbb{F}}$-vector space of solutions $\,\left\lbrace F\in {\mathbb{F}}[z_1,\ldots,z_c]: f\bullet F=0 \mbox{ for all }f\in I\right\rbrace$.

Inverse systems furnish an explicit bijection between items (b) and (c) in Theorem \[thm:main\]: $$\label{eq:HilbBijection}
\begin{array}{ccc}
\left\lbrace\begin{array}{c}
\mbox{points in }{\rm Hilb}^m\left({\mathbb{F}}[[\partial_{z_1},\ldots,\partial_{z_c}]]\right)\\
\end{array}\right\rbrace
& \longleftrightarrow &
\left\lbrace\begin{array}{c}
\mbox{$m$-dimensional ${\mathbb{F}}$-subspaces} \\ 
\mbox{of $ {\mathbb{F}}[z_1,\dots,z_c]$} \\ \mbox{closed under differentiation}
\end{array}\right\rbrace\\
I & \longrightarrow & V=I^\perp\\
I={\rm Ann}_{{\mathbb{F}}[[\partial_{z_1},\dots,\partial_{z_c}]]}(V) & \longleftarrow & V.\\
\end{array}$$

Setting $y_i=\partial_{z_i}$, the ideal in Example \[Bij:1,2\] is $I=\langle\partial_{z_2}^2,\partial_{z_1}\partial_{z_2},\partial_{z_1}^2+u_2^{\, -1}\partial_{z_2}\rangle\subset {\mathbb{F}}[[\partial_{z_1},\partial_{z_2}]]$. Note that $\,z_1^2-2u_2z_2\,$ belongs to the inverse system $I^\perp$ because this polynomial is annihilated by all operators in $I$. Applying the differential operators $\partial_{z_1}$ and $\partial_{z_1}^2$ to $B_3=z_1^2-2u_2z_2$ we obtain an ${\mathbb{F}}$-basis of the inverse system: $B_1=1$, $B_2=z_1$ and $B_3$. Moreover, $I^\perp$ is generated by $B_3$ as an ${\mathbb{F}}[[\partial_{z_1},\partial_{z_2}]]$-module. Hence $I$ is a Gorenstein ideal.

The correspondence between items (c) and (d) in Theorem \[thm:main\] links generators of the inverse system of $I$ with Noetherian operators for $Q$. These will be discussed in depth in Section \[sec3\]. Suppose we are given an ${\mathbb{F}}$-basis $\{B_1,\ldots,B_m\}$ of the inverse system $I^\perp$ in (c). After clearing denominators, we can write $B_i(\mathbf{u},\mathbf{z})=\sum_{\vert\alpha\vert\leq m}\lambda_\alpha(\mathbf{u})\mathbf{z}^\alpha$ where $\lambda_\alpha(\mathbf{u})$ is a polynomial in $R$ that represents a residue class modulo $P$. We now replace the unknown $z_i$ in these polynomials with the differential operator $\partial_{x_i}$. This gives the Noetherian operators $$\label{eq:resultingDO} \qquad
A_i(\mathbf{x},\partial_{x_1},\dots,\partial_{x_c})\,\,\,=\,\,\,\sum_{\vert\alpha\vert\leq m} \lambda_\alpha(\mathbf{x}) \partial_{x_1}^{\alpha_1}\cdots \partial_{x_c}^{\alpha_c} 
\qquad {\rm for} \,\,\, i =1,\ldots,m.$$ The transition from the $B_i$’s to the $A_i$’s is invertible, giving the bijection between (c) and (d).

\[ex:noeth\] Consider the ideal $Q$ in (\[eq:twistedcubic6\]) and $I$ in (\[eq:punctual2\]). From the generators $B_1(u,z)=1$, $B_2(u,z)=z_1$ and $B_3(u,z)=z_1^2-2u_2z_2$ of the inverse system $I^\perp$ in ${\mathbb{F}}[z_1,z_2]$, we obtain the three Noetherian operators $A_1=1$, $A_2=\partial_{x_1}$ and $A_3=\partial_{x_1}^2-2x_2\partial_{x_2}$ that encode $Q$. Note that $A_3$ alone does not determine $Q$, although $B_3$ is enough to generate the inverse system.

An Algebraic View on Ehrenpreis-Palamodov {#sec3}
=========================================

In this section we derive the Noetherian differential operators that are central to the Fundamental Principle of Ehrenpreis [@EHRENPREIS] and Palamodov [@PALAMODOV]. In particular, we present a practical algorithm that computes these operators for arbitrary primary ideals in a polynomial ring over a field ${\mathbb{K}}$ of characteristic zero. Our approach extends the algebraic theory in [@BRUMFIEL_DIFF_PRIM; @NOETH_OPS; @OBERST_NOETH_OPS] and the first algorithmic steps taken in [@DAMIANO; @STURMFELS_SOLVING]. For analytic aspects of the Ehrenpreis-Palamodov Theorem we refer to [@EHRENPREIS; @PALAMODOV] and to the books by Björk [@BJORK] and Hörmander [@HORMANDER].

Our point of departure is a prime ideal $P$ in the polynomial ring $R = {\mathbb{K}}[x_1,\ldots,x_n]$. We are interested in $P$-primary ideals. Later on we shall interpret these ideals as systems of linear PDE, by replacing each variable $x_i$ by a differential operator $ \partial_{z_i} = \partial / \partial z_i$. First, however, we take a different path, aimed to turn part (d) in Theorem \[thm:main\] into an algorithm.

After applying Noether normalization, $R/P$ is a finitely generated ${\mathbb{K}}[x_{c+1},\ldots,x_n$\]-module, where $c={\rm codim}(P)$. The relative Weyl algebra $D_{n,c} = {\mathbb{K}}\langle x_1,\ldots,x_n,\partial_{x_1},\ldots,\partial_{x_c} \rangle$ consists of linear differential operators with polynomial coefficients, where only derivatives for the first $c$ variables appear. Every operator $A = A(\mathbf{x}, \partial_\mathbf{x})$ in $D_{n,c}$ is a unique ${\mathbb{K}}$-linear combination of [*normal monomials*]{} $\,\mathbf{x}^\alpha \partial_\mathbf{x}^\beta = x_1^{\alpha_1} \cdots x_n^{\alpha_n} \partial_{x_1}^{\beta_1} \cdots \partial_{x_c}^{\beta_c}$, where $\alpha \in \mathbb{N}^n$, $\beta \in {\mathbb{N}}^c$. We write $A \bullet f$ for the natural action of $D_{n,c}$ on polynomials $f  \in R$., which is defined by $$x_i \bullet f = x_i \cdot f \quad {\rm and} \quad \partial_{x_i} \bullet f = \partial f / \partial x_i .$$

Suppose we are given $A_1,\ldots,A_m$ in the relative Weyl algebra $D_{n,c}$. This specifies $$\label{eq:fromAtoQ}
    Q \,\,=\,\, \big\{\, f \in R \,: A_l \bullet f \in P \;\text{ for } \,
    l = 1,2,\ldots,m    \,\big\}.$$ The set $Q$ is a ${\mathbb{K}}$-vector space. However, in general, the subspace $Q$ is not an ideal in $R$.

\[ex:leftright\] Fix $n=m=2$, $P = \langle x_1,x_2 \rangle$ and $A_1 = \partial_{x_1}$. If $A_2 = \partial_{x_2}$ then $Q $ is the space of polynomials $f$ in ${\mathbb{K}}[x_1,x_2]$ such that $x_1$ and $x_2$ do not appear in the expansion of $f$. That space is not an ideal. However, if $A_2 = 1$ then the formula (\[eq:fromAtoQ\]) gives the ideal $\,Q = \langle x_1^2, x_2 \rangle$.

\[rem:contains\_power\] The space $Q$ always contains a power of $P$. Namely, if $k$ is the maximal order among the operators $A_i$ then $P^{k+1} \subseteq Q$. This follows from the product rule of calculus.

We next present a necessary and sufficient condition for $m$ operators in $D_{n,c}$ to specify a primary ideal via (\[eq:fromAtoQ\]). We abbreviate $S= {\mathbb{K}}(x_{c+1},\ldots,x_n)[x_1,\ldots,x_c]$. The point in (\[eq:leftequalsright\]) below is that the relative Weyl algebra $D_{n,c}$ is both a left $R$-module and a right $R$-module.

\[thm:leftequalsright\] The space $Q$ is a $P$-primary ideal in the polynomial ring $R$ if and only if $$\label{eq:leftequalsright}
\qquad A_i \cdot x_j\,\,\in \,\,
 S\cdot \{A_1,\ldots,A_m\} \,+\, PS \cdot D_{n,c} 
\qquad \text{for}\,\, \, i =1,\ldots,m \,\,\text{and} \,\,j =1,\ldots,n.$$

In Example \[ex:leftright\] with $\{A_1,A_2\} = \{\partial_{x_1},\partial_{x_2} \}$ we have $ R = S$. Here $Q$ is not an ideal, and (\[eq:leftequalsright\]) fails indeed for $i=j=1$. To see this, one checks that $  \partial_{x_1} x_1 \not\in R\cdot \{\partial_{x_1},\partial_{x_2}\} + \langle x_1,x_2 \rangle D_{2,2}$. It would be desirable to turn the criterion in Theorem \[thm:leftequalsright\] into a general practical algorithm.

Suppose (\[eq:leftequalsright\]) holds and let $f \in Q$. By hypothesis, there exist $\,h_1,\ldots,h_m \in S$ such that $\,A_i x_j  \,=\, \sum_{k=1}^m h_k \,A_k \,$ modulo $ PS\cdot D_{n,c}$. Since $A_k \bullet f \in P$, we see that $A_i \bullet (x_j f) = (A_i \,x_j) \bullet f$ lies in $P$ for all $i,j$, and hence $x_j f \in Q$. Thus, $Q$ is an ideal.

Next we show that $Q$ is $P$-primary, by the following direct argument. Let $f,g \in R$ such that $f \cdot g \in Q$ and $g \not\in Q$. We claim that $f \in P$. We select an operator $A$ of minimal order among those inside $S\cdot \{A_1,\ldots,A_m\} + PS \cdot D_{n,c}$ that satisfy $A \bullet g \not\in PS$. The element $\,A \bullet (fg) \,=\, f \cdot (A \bullet g) \,+\, (A f - f A) \bullet g \,$ lies in $PS$. The commutator $A f - f A$ is a differential operator of order smaller than that of $A$. By (\[eq:leftequalsright\]), it is inside $S\cdot \{A_1,\ldots,A_m\} + PS \cdot D_{n,c}$. This ensures that $(A f - f A) \bullet g$ is in $PS$. We conclude that $f \cdot (A \bullet g) \in PS$. But, we know that $A \bullet g$ is not in $PS$, and hence $f$ is in the prime ideal $P$. Remark \[rem:contains\_power\] ensures that $\sqrt{Q}$ contains $ P$. Our argument shows that $Q$ is primary with $\sqrt{Q} = P$. The if-direction follows.

For the only-if-direction we utilize the isomorphism in Remark \[rem\_isom\_restrict\_Weyl\_mod\] and Lemma \[lem\_prim\_ideal\_implies\_bimod\]. The condition (\[eq:leftequalsright\]) is equivalent to the bi-module condition in Lemma \[lem\_prim\_ideal\_implies\_bimod\].

The following result is the key algebraic ingredient in the Ehrenpreis-Palamodov theory.

\[thm\_Noeth\_ops\] For every $P$-primary ideal $Q$ of multiplicity $m$ over $P$, there exist operators $A_1,\ldots,A_m$ in the relative Weyl algebra $D_{n,c}$ such that (\[eq:fromAtoQ\]) holds.

Theorem \[thm\_Noeth\_ops\] follows from Theorem \[thm:main\], to be proved in the next three sections. Indeed, if we are given a $P$-primary ideal $Q$ of multiplicity $m$ over $P$, then $Q$ specifies an $m$-dimensional $R$-bi-module inside the ${\mathbb{F}}$-vector space ${\mathbb{F}}\otimes_R D_{n,c}$. We choose elements $A_1,\ldots,A_m$ in $D_{n,c}$ whose images form an ${\mathbb{F}}$-basis for that $R$-bi-module. These operators satisfy (\[eq:fromAtoQ\]).

Following Palamodov [@PALAMODOV], we call $A_1,\ldots,A_m$ the [*Noetherian operators*]{} that encode the primary ideal $Q$. It is an essential feature that these are linear differential operators with polynomial coefficients. Operators with constant coefficients do not suffice. In other words, the Weyl algebra is essential in describing primary ideals. This key point is due to Palamodov. It had been overlooked initially by Gröbner and Ehrenpreis. For instance, consider the ideal $Q$ for $n=4, m=3$ in the Introduction. Three Noetherian operators $A_1,A_2,A_3$ are given in (\[eq:twistedcubic6\]), and it is instructive to verify condition (\[eq:leftequalsright\]). Algorithms for passing back and forth between Noetherian operators and ideal generators of $Q$ will be presented later in this section.

Our problem is to solve a homogeneous system of linear PDE with constant coefficients. This is given by the generators of a primary ideal $Q$ in ${\mathbb{K}}[x_1,\ldots,x_n]$, where $x_j$ stands for the differential operator $\partial_{z_j} = \partial / \partial z_j$ with respect to a new unknown $z_j$. Our aim is to characterize all sufficiently differentiable functions $\psi(z_1,\ldots,z_n)$ that are solutions to these PDE. This characterization is the content of the Ehrenpreis-Palamodov Theorem, to be stated below. Note that, if we are given an arbitrary system $J \subset R$ of such PDE then we can reduce to the case discussed here by computing a primary decomposition of the ideal $J$.

For the analytic aspects that follow, we work over the field ${\mathbb{K}}= {\mathbb{C} }$ of complex numbers. Suppose $Q = \langle p_1,p_2,\ldots,p_r \rangle$, where $p_k = p_k(\mathbf{x})$. The PDE we need to solve take the form: $$\label{eq:mustsolvethis}
\qquad p_k(\partial_\mathbf{z})\bullet \psi(\mathbf{z})\,\, =\,\, 0 \qquad \text{ for }  k=1,2,\ldots,r.$$ Let $\mathcal{K} \subset {\mathbb{R}}^n$ be a compact convex set. We seek all functions $\psi(\mathbf{z})$ in $\, C^\infty(\mathcal{K})\,$ that satisfy (\[eq:mustsolvethis\]). Here we also use vector notation, namely $\mathbf{z} = (z_1,\ldots,z_n)$ and $\partial_\mathbf{z} = (\partial_{z_1},\ldots,\partial_{z_n})$. According to Theorem \[thm\_Noeth\_ops\], there exist Noetherian operators $A_1(\mathbf{x},\partial_\mathbf{x}),\ldots,A_m(\mathbf{x},\partial_\mathbf{x})$ which encode the primary ideal $Q$ in the sense of (\[eq:fromAtoQ\]). In symbols, $\,    A_l( \mathbf{x}, \partial_\mathbf{x}) \bullet f \in P\,$ for all $l$.

Each $A_l$ is an element in the relative Weyl algebra $D_{n,c}$, given as a unique ${\mathbb{C} }$-linear combination of normal monomials $\,\mathbf{x}^\alpha \partial_\mathbf{x}^\beta $. This is important since $D_{n,c}$ is non-commutative. We now replace $\partial_\mathbf{x}$ by $\mathbf{z}$ in the normal monomials. This results in commutative polynomials $$\label{eq:thisresults}
  B_l (\mathbf{x},\mathbf{z}) \,\, := \,\, 
 A_l(\mathbf{x},\partial_\mathbf{x})|_{\partial_{x_1} \mapsto z_1,\ldots,
 \partial_{x_c} \mapsto z_c}
 \qquad {\rm for} \quad l=1,2,\ldots,m .$$ We call $B_1,\ldots,B_m$ the [*Noetherian multipliers*]{} of the primary ideal $Q$. These are polynomial in $n+c$ variables, obtained by reinterpreting the Noetherian (differential) operators. Note that $B_1,\ldots,B_m$ span the inverse system in Theorem \[thm:main\] (c) when viewed inside ${\mathbb{F}}[z_1,\ldots,z_c]$.

The Noetherian operators and Noetherian multipliers in the Introduction are $$\label{eq:NoetMult}
\begin{matrix}
  A_1 \,=\, 1\,,\;\, A_2\,=\, \partial_{x_1}
\,\,\,{\rm and} \,\,\,A_3 \,=\, \partial_{x_1}^2 \,-\, 2 \,x_2\,\partial_{x_2}, \\
 B_1 \,=\, 1\,,\;\,\, B_2\,=\, z_1\,
\,\,\,{\rm and}\, \,\,\,B_3 \,=\, z_1^2 \,-\, 2 \,x_2\,z_2. \qquad
\end{matrix}$$ We note that this is consistent with (\[eq:twistedcubic5\]) because $\,x_2 = st^2\,$ holds on the variety $\,V(P)$.

Here is now the celebrated result on solutions to linear PDE with constant coefficients:

\[thm:Palamodov\_Ehrenpreis\] Fix the system (\[eq:mustsolvethis\]) of PDE given by the $P$-primary ideal $Q$. Any solution $\psi$ in $C^\infty(\mathcal{K})$ has an integral representation $$\label{eq:anysolution}
    \psi(\mathbf{z}) \,\,\,= \,\,\, \sum_{l=1}^m\,\int_{V(P)} \!\! B_l\left(\mathbf{x},\mathbf{z}\right) 
    \exp\left( \mathbf{x}^t \,\mathbf{z} \right) d\mu_l(\mathbf{x})$$ for suitable measures $\mu_l$ supported in $V(P)$. And, conversely, all such functions are solutions.

We follow the conventions used in analysis (cf. [@BJORK Chapter 8]) and we write our system in terms of the differential operators $D_{z_j} = -i\partial_{z_j}$, where $i=\sqrt{-1}$. We can account for this in the Noetherian multipliers by replacing $\mathbf{x}$ with $-i \mathbf{x}$. It is shown in [@BJORK Theorem 1.3, page 339] that any solution in $C^\infty(\mathcal{K})$ to the system (\[eq:mustsolvethis\]) can be written as $$\psi(\mathbf{z}) \,\,=\,\, \sum_{l=1}^m\int_{V(P)} B_l\left(-i\mathbf{x},\mathbf{z}\right)
         \exp\left(-i \mathbf{x}^t \, \mathbf{z} \right) d\mu_l(\mathbf{x}).$$ We can now change variables, by incorporating the multiplication with $-i$ into the measures, to get the formula (\[eq:anysolution\]). Conversely, to see that any such integral $\psi(\mathbf{z})$ is a solution to the PDE (\[eq:mustsolvethis\]) given by $Q$, we differentiate under the integral sign and use the Fourier transform.

Consider the system of PDE determined by the ideal $Q$ in the Introduction. The Noetherian multipliers in (\[eq:NoetMult\]) furnish integral representations for all of its solutions: $$\psi(\mathbf{z}) \,\,= \,\,
   \int_{V(P)} \!\!\!\! \exp\left(\mathbf{x}^t \mathbf{z}\right) d\mu_1(\mathbf{x}) \,+\,
   \int_{V(P)}\!\!\!\!  z_1 \exp\left(\mathbf{x}^t \mathbf{z}\right) d\mu_2(\mathbf{x}) \, +\,
   \int_{V(P)} \!\!\!\! (z_1^2-2x_2 z_2) \exp\left(\mathbf{x}^t \mathbf{z}\right) d\mu_3(\mathbf{x}).$$ Here $\mu_1,\mu_2,\mu_3$ are measures supported on the variety $\,V(P) = \bigl\{ \,(s^2t, s t^2,s^3, t^3) \,:\, s,t \in {\mathbb{C} }\,\bigr\}$. The assertion in (\[eq:twistedcubic5\]) is obtained by pulling the integrals back to the $(s,t)$-plane via the parametrization of $V(P)$. This replaces the measures $\mu_i$ by their pull-backs to that plane.

We next present two algorithms for Theorem \[thm\_Noeth\_ops\]. The first is for computing Noetherian operators from the generators of $Q$, and the second for going in the reverse direction. A key ingredient is the map $\gamma$ in (\[eq\_map\_gamma\]) which we encode in the ideal $$\label{eq:weencode}
 \big\langle \,x_1-y_1-u_1,\,\ldots\,,\,x_c-y_c-u_c\,,\,\, x_{c+1}-u_{c+1}\,,
 \,\ldots\,,\,x_n-u_n \,\bigr\rangle.$$ This technique was used for encoding the differential operators in our running example in (\[eq:magic\]).

\[alg:forward\]\
[Input:]{} Generators $p_1,p_2,\ldots,p_r$ of a $P$-primary ideal $Q$ in $R ={\mathbb{K}}[x_1,\ldots,x_n]$.\
[Output:]{} Elements $A_1,A_2,\ldots,A_m $ in the relative Weyl algebra $D_{n,c}$ that satisfy (\[eq:fromAtoQ\]).\
1. Compute polynomials in ${\mathbb{F}}[y_1,\ldots,y_c]$ that generate the zero-dimensional ideal $I$ in (\[eq:corr12\]).\
2. Using linear algebra over ${\mathbb{F}}$, compute a basis $\{B_1,\ldots,B_m\}$ for the inverse system $I^\perp$.\
3. Lift each $B_i(\mathbf{u},\mathbf{z})$ to obtain the Noetherian multipliers $B_i(\mathbf{x},\mathbf{z})$.\
4. Replace $\mathbf{z}$ by $\partial_\mathbf{x}$ to get the Noetherian operators $A_i(\mathbf{x},\partial_\mathbf{x}) $ in (\[eq:resultingDO\]).

\[alg:backward\]\
[Input:]{} Elements $A_1,A_2,\ldots,A_m $ in the relative Weyl algebra $D_{n,c}$ that satisfy (\[eq:leftequalsright\]).\
[Output:]{} Generators $p_1,p_2,\ldots,p_r$ of a $P$-primary ideal $Q$ that is defined as in (\[eq:fromAtoQ\]).\
1. In each $A_i(\mathbf{x},\partial_\mathbf{x})$ replace $\partial_\mathbf{x}$ by $\mathbf{z}$ to obtain the $m$ Noetherian multipliers $B_i(\mathbf{x},\mathbf{z})$ in (\[eq:thisresults\]).\
2. Replace $\mathbf{x}$ by $\mathbf{u}$ to obtain an ${\mathbb{F}}$-basis $\{B_1,\ldots,B_m\}$ for the inverse system $I^\perp$.\
3. Using $\,{\mathbb{F}}$-linear algebra in ${\mathbb{F}}[y_1,\ldots,y_c]$, find generators for the zero-dimensional ideal $I$.\
4. Add the ideal $I$ to (\[eq:weencode\]) and eliminate $\{y_1,\ldots,y_c,u_1,\ldots,u_n\}$ to obtain generators of $Q$.

We implemented both of these algorithms in [Macaulay2]{}. The code is made available at <https://software.mis.mpg.de>. We hope to develop this further into a [Macaulay2]{} package.

We close this section by presenting a new example that explains the algorithms.

To illustrate Algorithm \[alg:forward\], let $n = 4$ and fix the prime $P = \langle x_1,x_2,x_3 \rangle$ that defines a line in $4$-space ${\mathbb{K}}^4$. The following ideal is $P$-primary of multiplicity $m = 4$: $$Q \,\,=\,\, \bigl\langle\, x_1^2, \,x_1 x_2,\, x_1 x_3, \,x_1 x_4-x_3^2+x_1, \,
    x_3^2 x_4-x_2^2, \,x_3^2 x_4-x_3^2-x_2 x_3+2 x_1 \,\bigr\rangle .$$ In Step 1 we replace $x_1,x_2,x_3$ by $y_1,y_2,y_3$ and $x_4$ by $u_4$ to get a zero-dimensional ideal $I$ in ${\mathbb{F}}[y_1,y_2,y_3]$, where ${\mathbb{F}}= {\mathbb{K}}(u_4)$. Note that $I$ contains $\langle y_1,y_2,y_3 \rangle^4$. To check that $I$ is a point in ${\rm Hilb}^4({\mathbb{F}}[[y_1,y_2,y_3]])$, we exhibit a flat deformation to the square of the maximal ideal: $$I \,\,=\,\,\bigl\langle\, y_1^2\,,\,y_1 y_2\,,\, y_1 y_3\, ,\,
    y_2^2 -(u_4^2+u_4) \,y_1\,,\,
    y_2 y_3 - (u_4^2 + 1) \,y_1\,,\,
    y_3^2 - (u_4+1)\, y_1\,\bigr\rangle.$$ The inverse system $I^\perp$ lives in ${\mathbb{F}}[z_1,z_2,z_3]$. It is the $4$-dimensional ${\mathbb{F}}$-vector space with basis $$B_1 \,=\,  (u_4^2+u_4) z_2^2 + 2 (u_4^2+1)z_2 z_3 + (u_4+1) z_3^2 +2z_1\,,\,
    B_2 =  z_2\,,\, B_3 = z_3\,, \,B_4 = 1.$$ Note that this space is closed under differentiation. The Noetherian operators in Step 4 are $$A_1 \,=\,
    (x_4^2+x_4) \partial_{x_2}^2 + 2 (x_4^2+1)
    \partial_{x_2} \partial_{x_3} + (x_4+1) \partial_{x_3}^2 +2 \partial_{x_1},\,\,
    A_2 = \partial_{x_2\,}, \,\, A_3 = \partial_{x_3} \,, \,\, A_4 = 1 .$$ We can now check that these four operators in $D_{4,3}$ represent the given primary ideal: $$Q  \,\,= \,\,\bigl\{\,f \in {\mathbb{K}}[x_1,x_2,x_3,x_4]:\, A_i \bullet f\in\langle x_1,x_2,x_3\rangle 
    \hbox{ for $i=1,2,3,4$} \,\bigr\} .$$

Reversing this entire computation is the point of Algorithm \[alg:backward\]. Starting from the operators $A_1,A_2,A_3,A_4$, we compute the polynomials $B_1,B_2,B_3,B_4$ in ${\mathbb{F}}[z_1,z_2,z_3]$, which span the inverse system $I^\perp$. In Step 3, we find generators of the ideal $I$ in ${\mathbb{F}}[y_1,y_2,y_3$\]. And, finally, from this one obtains generators of $Q$ by the elimination process described in Step 4.

Hilbert Schemes and Inverse Systems {#sec4}
===================================

In this section we provide a proof of the bijections between parts (a), (b) and (c) of Theorem \[thm:main\]. Here the key players are punctual Hilbert schemes and Macaulay’s inverse systems.

We retain the notation from Sections \[sec2\] and \[sec3\], and we write ${\mathfrak{p}}=PS$ for the extension of our prime ideal $P$ in $R = {\mathbb{K}}[x_1,\ldots,x_c,x_{c+1},\ldots,x_n]$ to $S={\mathbb{K}}(x_{c+1},\ldots,x_n)[x_1,\ldots,x_c]$. By Noether Normalization, we assume that ${\mathbb{K}}[x_{c+1},\ldots,x_n] \hookrightarrow R/P$ is an integral extension, and this implies that ${\mathfrak{p}}$ is a maximal ideal in $S$. Our first goal is to parametrize $P$-primary ideals of fixed multiplicity $m$ over $P$ by the punctual Hilbert scheme ${\rm Hilb}^m\bigl({\mathbb{F}}[[y_1,\ldots,y_c]]\bigr)$. A special role is played by the inclusion map $\gamma:R \hookrightarrow {\mathbb{F}}[y_1,\dots,y_c]$ in (\[eq\_map\_gamma\]). This induces an inclusion $\,\gamma_S : S \hookrightarrow {\mathbb{F}}[y_1,\ldots,y_c]$, also given by $\,x_i \mapsto y_i+u_i $ for $i \leq c\,$ and $\,x_j \mapsto u_j$ for $j > c$.

\[rem\_local\_primary\_ideals\] Since ${\mathbb{K}}[x_{c+1},\ldots,x_n] \cap P=0$, the canonical map $R \hookrightarrow S$ gives a bijection between $P$-primary ideals and ${\mathfrak{p}}$-primary ideals (see, e.g., [@MATSUMURA Theorem 4.1]).

The maximal irrelevant ideal in ${\mathbb{F}}[y_1,\dots,y_c]$ is denoted by ${\mathcal{M}}=\langle y_1,\ldots,y_c\rangle$. For any $f(\mathbf{x})=f(x_1,\ldots,x_n) \in P$, we have $f(\mathbf{u})=f(u_1,\ldots,u_n)=0$ in $ {\mathbb{F}}$. A Taylor expansion yields $$f(\mathbf{u}+\mathbf{y})
    \,\,=\,\,f(u_1+y_1,\ldots,u_c+y_c,u_{c+1},\ldots,u_n)
    \,\,=\,\, \sum_{\substack{\lambda \in {\mathbb{N}}^c\\\lvert\lambda \rvert > 0}} \frac{\partial^{\lvert\lambda\rvert} f}{\partial_{x_1}^{\lambda_1}\cdots\partial_{x_c}^{\lambda_c}}(\mathbf{u})\,\mathbf{y}^\lambda.$$ This shows that $\gamma(P) \subseteq {\mathcal{M}}$, and therefore $\gamma_S( {\mathfrak{p}}) \subseteq {\mathcal{M}}$. The next proposition establishes a bijection between ${\mathfrak{p}}$-primary ideals containing ${\mathfrak{p}}^m$ and ${\mathcal{M}}$-primary ideals containing ${\mathcal{M}}^m$.

\[prop\_corespondence\_primary\_ideals\] For all $m \ge 1$, the inclusion $\gamma_S$ induces the isomorphism of local rings $$S/{\mathfrak{p}}^m \,\,\xrightarrow{\cong}\,\, {\mathbb{F}}[y_1,\ldots,y_c]/{\mathcal{M}}^m.$$

This result has also appeared in [@BRUMFIEL_DIFF_PRIM Proposition 4.1] and [@NOETH_OPS Proposition 3.9]. In these sources it was assumed that ${\mathbb{K}}$ is a perfect field. This holds here since ${\rm char}({\mathbb{K}}) = 0$.

\[rem\_ident\_Hilb\] (i) Any ideal of colength $m$ in ${\mathbb{F}}[[y_1,\ldots,y_c]]$ contains the ideal $\langle y_1,\ldots,y_c \rangle^m $. Therefore, ${\rm Hilb}^m\bigl({\mathbb{F}}[[y_1,\ldots,y_c]]\bigr)$ can be identified with $\,{\rm Hilb}^m\bigl({\mathbb{F}}[[y_1,\ldots,y_c]]/ \langle y_1,\ldots,y_c \rangle^m \bigr).    $ (ii) Any $ \langle y_1,\ldots,y_c\rangle$-primary ideal of colength $m$ in the polynomial ring ${\mathbb{F}}[y_1,\ldots,y_c]$ contains the ideal $\langle y_1,\ldots,y_c \rangle^m \subset {\mathbb{F}}[y_1,\ldots,y_c]$. For all $m>0$, we have the natural isomorphism $$\frac{{\mathbb{F}}[[y_1,\ldots,y_c]]}{{ \langle y_1,\ldots,y_c \rangle}^m}
    \,\,\cong \,\,\frac{{\mathbb{F}}[y_1,\ldots,y_c]}{{ \langle y_1,\ldots,y_c \rangle }^m}.$$ Therefore, the $\langle y_1,\ldots,y_c \rangle$-primary ideals of colength $m$ in ${\mathbb{F}}[y_1,\ldots,y_c]$ are parametrized by ${\rm Hilb}^m \bigl( {\mathbb{F}}[[y_1,\ldots,y_c]] \bigr)$. From now on, $\langle y_1,\ldots,y_c \rangle $-primary ideals in the polynomial ring ${\mathbb{F}}[y_1,\ldots,y_c]$ will automatically be identified with ideals in the power series ring ${\mathbb{F}}[[y_1,\ldots,y_c]]$.

Now we are ready to prove the correspondence between parts (a) and (b) in Theorem \[thm:main\].

\[thm:param\_primary\] As asserted in (\[eq:corr12\]), there is a bijective correspondence $$\begin{array}{ccc}
    \left\lbrace\begin{array}{c}
        \mbox{$P$-primary ideals of $R$}\\
        \mbox{with multiplicity $m$ over $P$}
    \end{array}\right\rbrace
    
    & \longleftrightarrow &
    
    \left\lbrace\begin{array}{c}
        \mbox{points in }{\rm Hilb}^m({\mathbb{F}}[[y_1,\ldots,y_c]])\\
    \end{array}\right\rbrace\\
    
    Q & \longrightarrow & I=\langle y_1,\dots,y_c\rangle^m+\gamma(
    Q){\mathbb{F}}[y_1,\dots,y_c]\\
    Q=\gamma^{-1}(I) & \longleftarrow & I.
\end{array}$$

The canonical map $R \hookrightarrow S$ gives a bijection between $P$-primary ideals and ${\mathfrak{p}}$-primary ideals (Remark \[rem\_local\_primary\_ideals\]). Also, for any $P$-primary ideal $Q \subset R$ we have $R_P/QR_P \cong S_{{\mathfrak{p}}}/Q S_{{\mathfrak{p}}}$. So, nothing is changed if we take $S$ and ${\mathfrak{p}}$ instead of $R$ and $P$. We have the commutative diagram

\(m) \[matrix of math nodes,row sep=3em,column sep=9em,minimum width=1.7em, text height=1.5ex, text depth=0.25ex\] [ S & \[\]\
S/\^[m]{} & \[\]/\^[m]{}.\
]{}; (m-1-1) edge node \[above\] [$\gamma_S$]{} (m-1-2) (m-2-1) edge node \[above\] [$\cong$]{} (m-2-2) ; (m-1-1) – (m-2-1); (m-1-2) – (m-2-2);

The map in the bottom row is the isomorphism in Proposition \[prop\_corespondence\_primary\_ideals\]. This gives an inclusion-preserving bijection between ${\mathfrak{p}}$-primary ideals containing ${\mathfrak{p}}^m$ and ${\mathcal{M}}$-primary ideals containing ${\mathcal{M}}^{m}$, in particular, colength does not change under this correspondence. In explicit terms, the ${\mathcal{M}}$-primary ideal $I$ corresponding to a ${\mathfrak{p}}$-primary ideal $QS \supseteq {\mathfrak{p}}^m$ is $$I \,\,=\,\, {\mathcal{M}}^{m} \,+\,  \gamma_S(QS)\big({\mathbb{F}}[\mathbf{y}]\big).$$ And, the ${\mathfrak{p}}$-primary ideal $QS$ corresponding to an ${\mathcal{M}}$-primary ideal $I \supseteq {\mathcal{M}}^m$ is given by $$QS \,\,=\,\, \gamma_S^{-1}(I).$$ Finally, the result now follows from Remark \[rem\_ident\_Hilb\].

We next show the correspondence between parts (b) and (c) in Theorem \[thm:main\]. This follows from the usual Macaulay duality. Although this argument is well-known, we will need a short discussion to later connect parts (c) and (d) of Theorem \[thm:main\]. Consider the injective hull $E=E_{{\mathbb{F}}[[y_1,\ldots,y_c]]}({\mathbb{F}})$ of the residue field ${\mathbb{F}}\cong {\mathbb{F}}[[y_1,\ldots,y_c]]/\langle y_1,\ldots,y_c \rangle$ of ${\mathbb{F}}[[y_1,\ldots,y_c]]$. Since ${\mathbb{F}}[[y_1,\ldots,y_c]]$ is a formal power series ring, this equals the module of inverse polynomials: $$\label{eq_isom_E_inv_sys}
    E \,\,\cong \,\, {\mathbb{F}}[y_1^{-1},\ldots,y_c^{-1}].$$ For a derivation see e.g. [@Brodmann_Sharp_local_cohom Lemma 11.2.3, Example 13.5.3] or [@BRUNS_HERZOG Theorem 3.5.8].

Consider the polynomial ring ${\mathbb{F}}[z_1,\ldots,z_c]$ as an ${\mathbb{F}}[[y_1,\ldots,y_c]]$-module by setting that $y_i$ acts on ${\mathbb{F}}[z_1,\ldots,z_c]$ as $\partial_{z_i}$, that is, $y_i \cdot F = \partial_{z_i} \bullet F$ for any $F \in {\mathbb{F}}[z_1,\ldots,z_c]$. Since the field ${\mathbb{F}}$ has characteristic zero, we have the following isomorphism of ${\mathbb{F}}[[y_1,\ldots,y_c]]$-modules $$\label{eq_isom_inv_sys_char_zero}
    {\mathbb{F}}[y_1^{-1},\ldots,y_c^{-1}] \;\xrightarrow{\cong}\; {\mathbb{F}}[z_1,\ldots,z_c], \quad \frac{1}{\mathbf{y}^{\alpha}} \;\mapsto\;  \frac{\mathbf{z}^\alpha}{\alpha!}.$$

Now, Macaulay’s duality is simply performed via Matlis duality. We use ${\left(-\right)}^\vee$ to denote Matlis dual ${\left(-\right)}^\vee={{\normalfont\text{Hom}}}_{{\mathbb{F}}[[y_1,\ldots,y_c]]}\left(-,E\right)$. This is a contravariant exact functor which establishes an anti-equivalence between the full-subcategories of artinian ${\mathbb{F}}[[y_1,\ldots,y_c]]$-modules and finitely generated ${\mathbb{F}}[[y_1,\ldots,y_c]]$-modules (see, e.g., [@BRUNS_HERZOG Theorem 3.2.13]).

For any zero-dimensional ideal $I $ in the power series ring $
 {\mathbb{F}}[[y_1,\ldots,y_c]]$, the isomorphisms (\[eq\_isom\_E\_inv\_sys\]) and (\[eq\_isom\_inv\_sys\_char\_zero\]) together with Matlis duality yield the following identifications: $$I^{\perp} \,\,=\,\, \left\lbrace F\in {\mathbb{F}}[z_1,\ldots,z_c]: f\bullet F=0 \mbox{ for all }
f\in I\right\rbrace \,\,\cong \,\, (0 :_{E} I) \,\,\cong \,\, \bigl({\mathbb{F}}[[y_1,\ldots,y_c]]/I \bigr)^\vee.$$ On the other hand, consider any ${\mathbb{F}}[[y_1,\ldots,y_c]]$-submodule $V$ of $ {\mathbb{F}}[z_1,\ldots,z_c] \cong E$. Then $V$ is an ${\mathbb{F}}$-subspace of ${\mathbb{F}}[z_1,\ldots,z_c]$ that is closed by differentiation, as $y_i$ is identified with the operator $\partial_{z_i}$. Again, the isomorphisms (\[eq\_isom\_E\_inv\_sys\]) and (\[eq\_isom\_inv\_sys\_char\_zero\]) with Matlis duality give identifications $${\rm Ann}_{{\mathbb{F}}[[\partial_{z_1},\dots,\partial_{z_c}]]}(V) \,\, \cong \,\, 
{\rm Ann}_{{\mathbb{F}}[[y_1,\ldots,y_c]]}(V) \, \, \cong \,\,
 {\bigl( E /V \bigr)}^{\vee} \, \,\subset \,\, {\mathbb{F}}[[y_1,\ldots,y_c]].$$ Hence, from the above discussions, we get the connection between (b) and (c) in Theorem \[thm:main\].

\[thm:Macaulay\_dual\] As asserted in (\[eq:HilbBijection\]), there is a bijective correspondence $$\begin{array}{ccc}
    \left\lbrace\begin{array}{c}
    \mbox{points in }{\rm Hilb}^m\left({\mathbb{F}}[[\partial_{z_1},\ldots,\partial_{z_c}]]\right)\\
    \end{array}\right\rbrace
    
    & \longleftrightarrow &
    
    \left\lbrace\begin{array}{c}
        \mbox{$m$-dimensional ${\mathbb{F}}$-subspaces of}\\
        \mbox{${\mathbb{F}}[z_1,\dots,z_c]$ closed by differentiation}
    \end{array}\right\rbrace\\
    
    
    I & \longrightarrow & V=I^\perp\\
    I={\rm Ann}_{{\mathbb{F}}[[\partial_{z_1},\dots,\partial_{z_c}]]}(V) & \longleftarrow & V.\\
\end{array}$$

Differential Operators Revisited {#sec5}
================================

In this section we review basic material on differential operators in commutative algebra. This is used in Section \[sec6\] to complete the proof of Theorem \[thm:main\]. Even though the Noetherian operators $A_i$ live in the Weyl algebra, we need the abstract perspective to link them to the Weyl-Noether module (\[eq:relativeweyl\]). As before, ${\mathbb{K}}$ is a field of characteristic zero and $R={\mathbb{K}}[x_1,\ldots,x_n]$.

For two $R$-modules $M$ and $N$, we regard ${{\normalfont\text{Hom}}}_{\mathbb{K}}(M, N)$ as an $(R\otimes_{\mathbb{K}}R)$-module, by setting $$\left((r \otimes_{\mathbb{K}}s) \delta\right)(w)\,\, =\,\, r \delta(sw) \quad \text{ for all }\,\, \delta \in {{\normalfont\text{Hom}}}_{\mathbb{K}}(M, N), \; w \in M,\; r,s \in R.$$ This is equivalent to saying that ${{\normalfont\text{Hom}}}_{\mathbb{K}}(M,N)$ is an $R$-bi-module, where the action on the left is given by post-composing $(r \cdot \delta)(w)=r\delta(w)$ and the action on the right is given by pre-composing $(\delta \cdot s)(w)=\delta(sw)$, for all $\delta \in {{\normalfont\text{Hom}}}_{\mathbb{K}}(M, N), \; w \in M,\; r,s \in R$. We use the bracket notation $[\delta,r](w) = \delta(rw)-r\delta(w)$ for all $\delta \in {{\normalfont\text{Hom}}}_{\mathbb{K}}(M, N)$, $r \in R$ and $w \in M$.

\[nota\_T\_mod\_struct\] We write $T=R \otimes_{\mathbb{K}}R = {\mathbb{K}}[x_1, \ldots, x_n, y_1, \ldots, y_n]$ as a polynomial ring in $2n$ variables, where $x_i$ represents $x_i \otimes_{\mathbb{K}}1$ and $y_i$ represents $1 \otimes_{\mathbb{K}}x_i - x_i \otimes_{\mathbb{K}}1$. The action of $T$ on ${{\normalfont\text{Hom}}}_{\mathbb{K}}(M,N)$ is thus given as follows. For all $\delta \in {{\normalfont\text{Hom}}}_{\mathbb{K}}(M, N)$ and $w \in M$, we have $$(x_i \cdot \delta) (w) = x_i \delta(w) \quad \text{ and } \quad  (y_i \cdot \delta) (w)  = \delta(x_i w) - x_i\delta(w) = \left[\delta,x_i\right](w) \qquad {\rm for} \,\,i=1,\ldots,n.$$

Any $T$-module is regarded as an $R$-module via the canonical map $R \hookrightarrow T, x_i  \mapsto x_i $. Thus, any $T$-module is given an $R$-module structure by using the left factor $R\otimes_{\mathbb{K}}1 \subset T = R \otimes_{\mathbb{K}}R$. The ${\mathbb{K}}$-linear differential operators form a $T$-submodule of ${{\normalfont\text{Hom}}}_{\mathbb{K}}(M, N)$, defined as follows.

\[def\_diff\_ops\] Let $M, N$ be $R$-modules. The *$m$-th order ${\mathbb{K}}$-linear differential operators ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(M, N) \subseteq {{\normalfont\text{Hom}}}_{\mathbb{K}}(M, N)$ from $M$ to $N$* form a $T$-module that is defined inductively by:

(i) ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{0}(M,N) \,:=\, {{\normalfont\text{Hom}}}_R(M,N)$.

(ii) ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m}(M, N) \,:= \,
            \big\lbrace \delta \in {{\normalfont\text{Hom}}}_{\mathbb{K}}(M,N) : \,[\delta, r] \in {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(M, N) 
            \,\text{ for all }\, r \in R \big\rbrace$.

The set of all *${\mathbb{K}}$-linear differential operators from $M$ to $N$* is the $T$-module $${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(M, N) \,\,:=\,\, \bigcup_{m=0}^\infty {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(M,N).$$ Subsets $\mathcal{E} \subseteq {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(M, N)$ are viewed as differential equations. Their solutions spaces are $$\label{eq:solE}
{\rm Sol}(\mathcal{E}) \,\,:= \,\,\big\lbrace w \in M : \delta(w) = 0 \text{ for all } \delta \in \mathcal{E} \big\rbrace 
\,\,= \,\,\bigcap_{\delta \in \mathcal{E} } {\rm Ker}(\delta).$$

Following the approach in [@NOETH_OPS Section 2], we now introduce the module of principal parts. By construction, the ideal $ \Delta_{R/{\mathbb{K}}} = \langle y_1, \ldots,y_n \rangle$ in $T$ is the kernel of the multiplication map $$T = R \otimes_{\mathbb{K}}R \,\rightarrow \,R\,, \quad     r \otimes_{\mathbb{K}}s \,\mapsto\, rs.$$

Let $M$ be an $R$-module. The module of *$m$-th principal parts of $M$* equals $$P_{R/{\mathbb{K}}}^m(M) \,\,:=\,\, \frac{R \otimes_{\mathbb{K}}M}{\Delta_{R/{\mathbb{K}}}^{m+1}  \left(R \otimes_{\mathbb{K}}M\right)}.$$ This is a $T$-module. It comes with the natural map $\,d^m : \,M \rightarrow P_{R/{\mathbb{K}}}^m(M),\,
w \mapsto \overline{1 \otimes_{\mathbb{K}}w}$. In the special case $M=R$ we abbreviate $\,P_{R/{\mathbb{K}}}^m \,:=\, P_{R/{\mathbb{K}}}^m(R)=T/\Delta_{R/{\mathbb{K}}}^{m+1}$, and the map becomes $$\label{eq_univ_diff}
    d^m : R \rightarrow P_{R/{\mathbb{K}}}^m, \;\;x_i \, \mapsto \,\overline{1 \otimes_{\mathbb{K}}x_i} \,=\, \overline{x_i+y_i} .$$

The following proposition offers a fundamental characterization of differential operators.

\[prop\_represen\_diff\_opp\] Let $m\ge 0$ and let $M, N$ be $R$-modules. Then, the following map is an isomorphism of $R$-modules: $$\begin{aligned}
    {\left(d^m\right)}^*\, :\, {{\normalfont\text{Hom}}}_R\left(P_{R/{\mathbb{K}}}^m(M), N\right) 
    &\,\,\xrightarrow{\cong} \,\,{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(M, N), \\
    \varphi  \quad &\,\,\mapsto \,\quad \varphi \circ d^m.
    \end{aligned}$$

This is a very general result for commutative rings $R$. What we are interested in here is the polynomial ring $R = {\mathbb{K}}[x_1,\ldots,x_n]$ over a field ${\mathbb{K}}$ of characteristic zero. In this case, the $R$-module $P_{R/{\mathbb{K}}}^m=T/\Delta_{R/{\mathbb{K}}}^{m+1}$ is free, and a basis is given by $\bf y$-monomials of degree at most $m$: $$\label{eq_direct_sum_Prin}
P_{R/{\mathbb{K}}}^m \,\,\,= \,\,\bigoplus_{\lvert \alpha \rvert \le m} R\mathbf{y}^\alpha \quad =
\bigoplus_{\alpha_1 + \cdots + \alpha_r \le m} \!\!\!\! R y_1^{\alpha_1}\cdots y_n^{\alpha_n}.$$ Proposition \[prop\_represen\_diff\_opp\] implies that $\,{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R, R)\, \cong\,
  {{\normalfont\text{Hom}}}_R\bigl(P_{R/{\mathbb{K}}}^m, R\bigr)\,$ is a free $R$-module with basis $$\label{eq_basis_diff_ops}
\big\{ {(y_1^{\alpha_1}\cdots y_n^{\alpha_n})}^* \circ d^m  : \alpha_1 + \cdots + \alpha_n \le m \big\}.$$ For any polynomial $f(\mathbf{x})$ in $ R$, the operator $d^m$ in (\[eq\_univ\_diff\]) computes the Taylor expansion $$d^m(f(\mathbf{x})) \,\,= \,\,f(1\otimes_{\mathbb{K}}\mathbf{x}) \,\,= \,\,f(\mathbf{x}+\mathbf{y}) \,\,=\,\,
\sum_{\lambda \in {\mathbb{N}}^n} \left(D_\mathbf{x}^\lambda f\right)\!(\mathbf{x})\,\mathbf{y}^\lambda,$$ where $\,D_\mathbf{x}^{\lambda}:R\rightarrow R\,$ is the differential operator we all know from calculus: $$D_\mathbf{x}^{\lambda}\,\, =\,\, \frac{1}{\lambda!}\partial_\mathbf{x}^\lambda \,\,= \,\,
\frac{1}{\lambda_1!\cdots \lambda_n!}\partial_{x^1}^{\lambda_1}\cdots \partial_{x^n}^{\lambda_n}.$$ For any $\alpha \in {\mathbb{N}}^n$ we thus have $\,
\left({(\mathbf{y}^\alpha)}^* \circ d^m\right)(f(\mathbf{x})) = \left(D_\mathbf{x}^\alpha f\right)(\mathbf{x})$. The equation (\[eq\_basis\_diff\_ops\]) implies $${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R)\,\, =\,\,\bigoplus_{\lvert \alpha \rvert \le m} R D_\mathbf{x}^\alpha
\,\,=\,\, \bigoplus_{\lvert \alpha \rvert \le m} R \partial_\mathbf{x}^\alpha.$$ By letting $m$ go to infinity, we now recover the Weyl algebra in its well-known role:

\[lem\_Weyl\_as\_diff\_ops\] $\,{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(R, R)$ coincides with the Weyl algebra ${\mathbb{K}}\langle x_1,\ldots,x_n,\partial_{x_1},\ldots,\partial_{x_n} \rangle$.

Let $J $ be an ideal in $R= {\mathbb{K}}[x_1,\ldots,x_n]$. The canonical projection $\pi : R \rightarrow R/J$ induces a natural map of differential operators. This is the following homomorphism of $T$-modules: $$\label{eq:pimap}
{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(\pi) \,: \,{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R, R) \,\rightarrow \,
{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R, R/J), \quad \delta \,\mapsto\, \pi \circ \delta.$$

\[lem\_diff\_ops\_R/J\] We have the following explicit description of the objects in (\[eq:pimap\]):

(i) ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R, R/J)$ is a free $R/J$-module with direct summands decomposition $$\qquad
            {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R/J) \,\,= \,\,\bigoplus_{\lvert \alpha \rvert \le m} (R/J) \overline{D_\mathbf{x}^\alpha}, \qquad {\rm where}\,\,\,
    \overline{D_\mathbf{x}^\alpha} = \pi \circ D_\mathbf{x}^\alpha.$$

(ii) The map ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(\pi)$ is surjective. Explicitly, any differential operator $$\epsilon \,\,=\, \sum_{\lvert \alpha \rvert \le m} \overline{r_\alpha}
     \overline{D_\mathbf{x}^\alpha} \,\in\, {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R/J), \quad \text{ where } r_\alpha \in R,$$ can be lifted to an operator $\,\delta=\sum_{\lvert \alpha \rvert \le m} r_\alpha D_\mathbf{x}^\alpha \,\in\, {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R)\,$ with $\,\epsilon = {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(\pi)(\delta)$.

$(i)$ From Proposition \[prop\_represen\_diff\_opp\] and the Hom-tensor adjunction we obtain the isomorphisms $$\begin{aligned}
    \label{eq_isoms_Diff_R/J}
    \begin{split}
    {{\normalfont\text{Hom}}}_{R/J}\left(R/J \otimes_R P_{R/{\mathbb{K}}}^m, R/J\right) &
    \,\,\cong \,\, {{\normalfont\text{Hom}}}_R\left(P_{R/{\mathbb{K}}}^m, {{\normalfont\text{Hom}}}_{R/J}\left(R/J, R/J\right)\right)\\
    & \,\,\cong \,\,{{\normalfont\text{Hom}}}_R\left(P_{R/{\mathbb{K}}}^m, R/J\right)\\
    &\,\,\cong \,\,{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R/J).
    \end{split}         
    \end{aligned}$$ The isomorphism from the first row to the second row in (\[eq\_isoms\_Diff\_R/J\]) is given by $$\psi \in {{\normalfont\text{Hom}}}_{R/J}\left(R/J \otimes_R P_{R/{\mathbb{K}}}^m, R/J\right) \;\;\mapsto \;\;\psi\circ h_m \in {{\normalfont\text{Hom}}}_R\left(P_{R/{\mathbb{K}}}^m, R/J\right),$$ where $\,h_m\,$ is the canonical map $\, P_{R/{\mathbb{K}}}^m \rightarrow R/J \otimes_R P_{R/{\mathbb{K}}}^m$. Therefore, the isomorphism from the first to the third row in (\[eq\_isoms\_Diff\_R/J\]) is given explicitly as $ \,\psi\, \mapsto \, \psi \circ h_m \circ d^m $. By using equation (\[eq\_direct\_sum\_Prin\]) we get that $R/J \otimes_R P_{R/{\mathbb{K}}}^m$ is a free $R/J$-module with direct summands decomposition $$R/J \otimes_R P_{R/{\mathbb{K}}}^m \,\,=
\,\, \bigoplus_{\lvert \alpha \rvert \le m} (R/J)\mathbf{y}^\alpha.$$ Our explicit isomorphism for (\[eq\_isoms\_Diff\_R/J\]) shows that ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R/J)$ is a free $R/J$-module with basis $$\big\{ {(y_1^{\alpha_1}\cdots y_n^{\alpha_n})}^* \circ h_m \circ d^m  : \alpha_1 + \cdots + \alpha_n \le m \big\}.$$ Now, for any polynomial $f(\mathbf{x})$ in $ R$, we obtain the equations $$\begin{aligned}
    \label{eq_diff_opp_z_alpha}     
    \begin{split}
    \left({(\mathbf{y}^\alpha)}^* \circ h_m \circ d^m\right)(f(\mathbf{x})) &\,\,
    = \,\,\left({(\mathbf{y}^\alpha)}^* \circ h_m\right)\left(\sum_{\lambda \in {\mathbb{N}}^n} \left(D_\mathbf{x}^\lambda f\right)(\mathbf{x})\mathbf{y}^\lambda\right)\\
    &\,\,=\,\, \left({(\mathbf{y}^\alpha)}^*\right)\left(\sum_{\lambda \in {\mathbb{N}}^n} \pi\left(\left(D_\mathbf{x}^\lambda f\right)(\mathbf{x})\right)\mathbf{y}^\lambda\right)
    \,\,=\,\,\,\pi\big(\left(D_\mathbf{x}^\alpha f\right)(\mathbf{x})\big).
    \end{split}     
    \end{aligned}$$ This implies that the operators $\overline{D_\mathbf{x}^\alpha}=\pi \circ D_\mathbf{x}^\alpha$ with $\lvert\alpha\rvert \le m$ give a basis of ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R, R/J)$. Part $(ii)$ follows directly from part $(i)$. This concludes the proof of Lemma \[lem\_diff\_ops\_R/J\].

Since $R$ is a polynomial ring, the process of lifting differential operators is easy and explicit. However, the surjectivity of ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(\pi)$ is a subtle property, and it is not always satisfied over more general types of rings. For an illustration see [@NOETH_OPS Example 5.2].

Proof of the Representation Theorem {#sec6}
===================================

We here finish the proof of Theorem \[thm:main\] by connecting part (d) with parts (a), (b), and (c). The section is divided into two subsections. In the first one, we treat the zero-dimensional situation, where $c=n$. In the second one, we use Noether normalization and the results on differential operators in Section \[sec5\] to reduce the general case to the zero-dimensional case.

The zero-dimensional case
-------------------------

We here restrict ourselves to ideals in $R= {\mathbb{K}}[x_1,\ldots,x_n]$ that are primary to a maximal ideal $P$. Hence $c=n$ and ${\mathbb{F}}=R/P$. Since the base field ${\mathbb{K}}$ is assumed to have characteristic zero, an adaptation of Gröbner’s classical approach via Macaulay’s inverse system will be valid.

Using the notation $\,T=R\otimes_{\mathbb{K}}R={\mathbb{K}}[x_1,\ldots,x_n,y_1,\ldots,y_n]\,$ from Section \[sec5\], we now have $$\label{eq:FFRT}
{\mathbb{F}}\otimes_R T \,\,=\,\, {\mathbb{F}}\otimes_R \left(R\otimes_{\mathbb{K}}R\right) \,\,\cong\,\,
 R/P \otimes_{\mathbb{K}}{\mathbb{K}}[y_1,\ldots,y_n] \,\,\cong\,\, {\mathbb{F}}[y_1, \ldots, y_n].$$ This endows ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m}\left(R, {\mathbb{F}}\right)$ with the structure of an ${\mathbb{F}}[y_1,\ldots,y_n]$-module. Applying Lemma \[lem\_diff\_ops\_R/J\] with $J=P$, we see that ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m}\left(R, {\mathbb{F}}\right)$ is a finite-dimensional ${\mathbb{F}}$-vector space. In the sequel, the irrelevant maximal ideal $
{\mathcal{M}}= \langle y_1,\ldots,y_n \rangle \subset {\mathbb{F}}[y_1,\ldots,y_n]
$ will play an important role. This ideal is also given as ${\mathcal{M}}=\Delta_{R/{\mathbb{K}}} \left( {\mathbb{F}}\otimes_R T \right)$. For any $m \ge 0$ we identify $$\frac{{\mathbb{F}}[y_1,\ldots,y_n]}{{\mathcal{M}}^{m+1}} \,\,\,=\,\,
 \bigoplus_{\vert\alpha\rvert \le m} {\mathbb{F}}\mathbf{y}^{\alpha}.$$ For any ${\mathbb{F}}[y_1,\ldots,y_n]$-module $M$, the ${\mathbb{F}}$-dual ${{\normalfont\text{Hom}}}_{{\mathbb{F}}}(M, {\mathbb{F}})$ is naturally an ${\mathbb{F}}[y_1,\ldots,y_n]$-module as follows: if $\psi \in {{\normalfont\text{Hom}}}_{{\mathbb{F}}}(M, {\mathbb{F}})$, then $y_i \cdot \psi$ is the ${\mathbb{F}}$-linear map $\,\psi(y_i \cdot -)  :  w \in M \mapsto \psi(y_iw) \in {\mathbb{F}}$.

The next result relates submodules of ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m}\left(R, {\mathbb{F}}\right)$ to ${\mathcal{M}}$-primary ideals in ${\mathbb{F}}[y_1,\ldots,y_n]$.

\[lem\_descrip\_diff\_opp\] The following statements hold for all positive integers $m$:

(i) We have an isomorphism of ${\mathbb{F}}[y_1,\ldots,y_n]$-modules $${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}\left(R, {\mathbb{F}}\right)\,\, \cong \,\,
            {{\normalfont\text{Hom}}}_{{\mathbb{F}}}\Bigg(\frac{{\mathbb{F}}[y_1,\ldots,y_n]}{{{\mathcal{M}}}^{m}}, {\mathbb{F}}\Bigg).$$

(ii) The following map gives a bijective correspondence between ${\mathcal{M}}$-primary ideals $I $ in $ {\mathbb{F}}[y_1,\ldots,y_n]$ that contain ${\mathcal{M}}^{m}$ and ${\mathbb{F}}[y_1,\ldots,y_n]$-submodules of $\,{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}\left(R,{\mathbb{F}}\right)$: $$\label{eq:ImapHom}
    I  \;\mapsto\; {{\normalfont\text{Hom}}}_{{\mathbb{F}}}\left(\frac{{\mathbb{F}}[y_1,\ldots,y_n]}{I}, {\mathbb{F}}\right).$$

(iii) Let $\mathcal{E} \subseteq {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,{\mathbb{F}})$ be the image under (\[eq:ImapHom\]) of an ${\mathcal{M}}$-primary ideal $I \supseteq {\mathcal{M}}^{m}$. Then $${\rm Sol}(\mathcal{E}) \,\,=\,\, \gamma^{-1}(I),$$ with notation as in (\[eq:solE\]), where $\gamma$ is the inclusion $ R  \hookrightarrow {\mathbb{F}}[y_1,\ldots,y_n], x_i \mapsto y_i{+}u_i$ in (\[eq\_map\_gamma\]).

This is essentially [@NOETH_OPS Lemma 3.8]. We provide a proof for the sake of completeness.

$(i)$ Since ${\mathbb{F}}=R/P$, from equation (\[eq\_isoms\_Diff\_R/J\]) we obtain the isomorphism $${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R, {\mathbb{F}})\,\, \cong \,\,{{\normalfont\text{Hom}}}_{{\mathbb{F}}}\bigl({\mathbb{F}}\otimes_R P_{R/{\mathbb{K}}}^{m-1}, {\mathbb{F}}\bigr).$$ Thus, the result follows from the fact that $\,{\mathbb{F}}\otimes_R P_{R/{\mathbb{K}}}^{m-1} \,\cong\, {\mathbb{F}}[\mathbf{y}]/ {\mathcal{M}}^{m} $.

$(ii)$ Since ${\mathbb{F}}[\mathbf{y}]/{\mathcal{M}}^{m}$ is a finite-dimensional vector space over ${\mathbb{F}}$, the functor ${{\normalfont\text{Hom}}}_{{\mathbb{F}}}\left(-,{\mathbb{F}}\right)$ gives a bijection between quotients of ${\mathbb{F}}[\mathbf{y}]/{\mathcal{M}}^{m}$ and ${\mathbb{F}}[\mathbf{y}]$-submodules of ${{{\normalfont\text{Hom}}}_{{\mathbb{F}}}\left(\frac{{\mathbb{F}}[\mathbf{y}]}{{\mathcal{M}}^{m}},{\mathbb{F}}\right)}$. So, the claim follows from $(i)$.

$(iii)$ By assumption, $\,\mathcal{E} \,=\,{{{\normalfont\text{Hom}}}_{{\mathbb{F}}}\left(\frac{{\mathbb{F}}[\mathbf{y}]}{I},{\mathbb{F}}\right)} \,$ is in $\,{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,{\mathbb{F}})$. Consider the canonical map $ \,\Phi_I : \frac{{\mathbb{F}}[\mathbf{y}]}{{\mathcal{M}}^{m}}  \twoheadrightarrow \frac{{\mathbb{F}}[\mathbf{y}]}{I}\,$ given by the $\mathcal{M}$-primary ideal $I \supseteq {\mathcal{M}}^{m}$. From the isomorphism (\[eq\_isoms\_Diff\_R/J\]) we get $${\rm Sol}(\mathcal{E})  \,\,=\,\,
 \bigl\{ f \in R : \left(\psi \circ \Phi_I \circ h_{m-1} \circ d^{m-1}\right)(f)=0 
 \text{ for all } \psi \in {{{\normalfont\text{Hom}}}_{{\mathbb{F}}}\left({\mathbb{F}}[\mathbf{y}]/I ,{\mathbb{F}}\right)} \bigr\}.$$ The composition $\Phi_I \circ h_{m-1} \circ d^{m-1}$ coincides with the map $\,R \mapsto {\mathbb{F}}[\mathbf{y}]/ I,\, x_i \mapsto \overline{y_i+u_i}$. Hence $$\begin{aligned}
        {\rm Sol}(\mathcal{E}) &\,\,=\,\, \bigl\{\,f \in R : 
        \psi\bigl(\,\overline{f(
        \mathbf{y}+\mathbf{u}})\, \bigr)=0\, \text{ for all } \,\psi \in {{{\normalfont\text{Hom}}}_{{\mathbb{F}}}\left({\mathbb{F}}[\mathbf{y}]/ I ,{\mathbb{F}}\right)}\bigr\} \\
        &\,\,=\,\, \bigl\{\, f \in R : f\bigl(\mathbf{y}+\mathbf{u}\bigr) \in I \, \bigr\}
        \,\,=\,\, \gamma^{-1}(I).
        \end{aligned}$$ This completes the proof of Proposition \[lem\_descrip\_diff\_opp\].

Next, under the assumption of $P$ being maximal, we relate part (d) with the other parts in Theorem \[thm:main\]. By Definition \[def\_diff\_ops\] and Lemma \[lem\_Weyl\_as\_diff\_ops\], the Weyl-Noether module has the filtration $${\mathbb{F}}\,\otimes_R \, R\langle\partial_{x_1},\ldots,\partial_{x_n}\rangle \,\;= \;\, {\mathbb{F}}\, \otimes_R \, \biggl(\lim\limits_{\substack{\longrightarrow\\m}} {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R) \biggr) \,\;\cong \; \,\lim\limits_{\substack{\longrightarrow\\m}}\Big({\mathbb{F}}\otimes_R {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R)\Big).$$ Applying Lemma \[lem\_diff\_ops\_R/J\] with $J=P$ gives ${\mathbb{F}}\otimes_R {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R) \cong {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,{\mathbb{F}}) \cong \bigoplus_{\vert\alpha\rvert\le m}{\mathbb{F}}\overline{\partial_{\mathbf{x}}^\alpha}$. This gives rise to the following isomorphisms of ${\mathbb{F}}$-vector spaces: $$\label{eq_isom_relWeyl_diff}
    {\mathbb{F}}\,\otimes_R \, R\langle\partial_{x_1},\ldots,\partial_{x_n}\rangle \;\cong \; {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(R,{\mathbb{F}}) \;\cong \; \bigoplus_{ \alpha  \in {\mathbb{N}}^n} {\mathbb{F}}\overline{\partial_\mathbf{x}^\alpha}.$$ When the Weyl-Noether module was introduced in (\[eq:relativeweyl\]), we gave a purely algebro-symbolic treatment and we noticed that an ${\mathbb{F}}$-basis is given by $\,\left\lbrace 1 \otimes_R \partial_{\bf x}^\alpha: \alpha \in{\mathbb{N}}^n\right\rbrace$. Now, with the isomorphism (\[eq\_isom\_relWeyl\_diff\]), the elements $1 \otimes_R \partial_{\mathbf{x}}^\alpha $ are seen as the differential operators $\overline{\partial_{\mathbf{x}}^\alpha} \in {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(R, {\mathbb{F}})$.

The following map is an isomorphism of ${\mathbb{F}}$-vector spaces: $$\label{eq_map_omega}
\omega : {\mathbb{F}}[z_1,\ldots,z_n] \;\rightarrow\; {\mathbb{F}}\,\otimes_R \,  R\langle\partial_{x_1},\ldots,\partial_{x_n}\rangle\;\cong\;{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(R,{\mathbb{F}}), \quad \mathbf{z}^\alpha \mapsto \partial_{\mathbf{x}}^\alpha.$$ From (\[eq:FFRT\]) and Notation \[nota\_T\_mod\_struct\] we get the following actions. For any $\alpha \in {\mathbb{N}}^n$ and $1 \le i \le n$, $$\label{eq_deriv_z_bracket_partial}
    \partial_{z_i} \bullet \mathbf{z}^\alpha \,=\, \alpha_iz_1^{\alpha_1}\cdots z_i^{\alpha_i-1}\cdots z_n^{\alpha_n} \quad\text{ and }\quad y_i\cdot\partial_{\mathbf{x}}^\alpha
    \,=\,\left[\partial_{\mathbf{x}}^\alpha,x_i\right]\,=\,
\alpha_i\partial_{x_1}^{\alpha_1}\cdots \partial_{x_i}^{\alpha_i-1}\cdots \partial_{x_n}^{\alpha_n}.$$ Hence the map $\omega$ in (\[eq\_map\_omega\]) gives a bijection between ${\mathbb{F}}$-vector subspaces of ${\mathbb{F}}[z_1,\ldots,z_n]$ closed under differentiation and ${\mathbb{F}}[y_1,\ldots,y_n]$-submodules of ${\mathbb{F}}\otimes_R  R\langle\partial_{x_1},\ldots,\partial_{x_n}\rangle$. The latter structure as a submodule is equivalent to being an $R$-bi-submodule of the Weyl-Noether module.

\[lem\_prim\_ideal\_implies\_bimod\] Let $\mathcal{E}$ be a finite dimensional ${\mathbb{F}}$-vector subspace of $ {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(R,{\mathbb{F}})$. If $\,Q = {\rm Sol}(\mathcal{E})\,$ is a $P$-primary ideal in $R = {\mathbb{K}}[x_1,\ldots,x_n]$ then $\mathcal{E}$ is an $R$-bi-module.

Fix $m \in {\mathbb{N}}$ such that $Q \supseteq P^m$ and $\mathcal{E} \subseteq {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,{\mathbb{F}})$. The map $\gamma$ in (\[eq\_map\_gamma\]) defines the ideal $I={\mathcal{M}}^m+\gamma(
Q){\mathbb{F}}[y_1,\dots,y_n]$. Let $\mathcal{E}^\prime \subseteq {{\normalfont\text{Hom}}}_{\mathbb{F}}\left(\frac{{\mathbb{F}}[\mathbf{y}]}{{\mathcal{M}}^m},{\mathbb{F}}\right)$ be the ${\mathbb{F}}$-vector subspace coming from $\mathcal{E}$ under the isomorphism of Proposition \[lem\_descrip\_diff\_opp\]$(i)$. The hypothesis $\,Q={\rm Sol}(\mathcal{E})\,$ implies $$\label{eq_functionals_sols}
I/{\mathcal{M}}^m\,\, = \,\,\bigl\{ \,w \in {\mathbb{F}}[\mathbf{y}]/ {\mathcal{M}}^m
 \,:\, \delta(w)=0 \,\text{ for all } \,\delta \in \mathcal{E}^\prime\, \bigr\}.$$ Dualizing the inclusion $\mathcal{E}^\prime \hookrightarrow
     {{\normalfont\text{Hom}}}_{\mathbb{F}}\left( {\mathbb{F}}[\mathbf{y}]/{\mathcal{M}}^m,{\mathbb{F}}\right)$ we get the short exact sequence $$\label{eq_dualize_short_E}
0 \,\,\rightarrow \,\,Z \,\,\rightarrow\,\, {\mathbb{F}}[\mathbf{y}] / {\mathcal{M}}^m
\,\, \rightarrow \,\,{{\normalfont\text{Hom}}}_{\mathbb{F}}(\mathcal{E}^\prime, {\mathbb{F}})\,\, \rightarrow \,\, 0,$$ where $Z=\Big\lbrace w \in \frac{{\mathbb{F}}[\mathbf{y}]}{{\mathcal{M}}^m} 
 \,:\, \delta(w)=0 \text{ for all } \delta \in \mathcal{E}^\prime\Big\rbrace$. Therefore, equations (\[eq\_functionals\_sols\]) and (\[eq\_dualize\_short\_E\]) yield the isomorphism $\,{{\normalfont\text{Hom}}}_{\mathbb{F}}(\mathcal{E}^\prime,{\mathbb{F}}) \cong {\mathbb{F}}[\mathbf{y}]/I$, and we conclude that $\mathcal{E} \cong \mathcal{E}^\prime$ is an $R$-bi-module.

Finally, to complete the proof of Theorem \[thm:main\], it will suffice to prove the following.

\[thm\_noeth\_ops\_zero\_dim\] Let $P$ be a maximal ideal in $R = {\mathbb{K}}[x_1,\ldots,x_n]$, and let $Q \subset R$ be a $P$-primary ideal of multiplicity $m$ over $P$. Then $\,Q = {\rm Sol}(\mathcal{E})$, where $\mathcal{E}$ is obtained by the following steps:

(i) As in Theorem \[thm:param\_primary\], set $\,I=\langle y_1,\dots,y_n\rangle^m+\gamma(
            Q){\mathbb{F}}[y_1,\dots,y_n]$.

(ii) As in Theorem \[thm:Macaulay\_dual\], set $\,V = I^\perp \,\subset\, {\mathbb{F}}[z_1,\ldots,z_n]$.

(iii) Using the map $\omega$ in (\[eq\_map\_omega\]), set $\,\,\mathcal{E}=\omega(V) \;\subset \;{\mathbb{F}}\otimes_R R\langle\partial_{x_1},\ldots,\partial_{x_n}\rangle\;\cong\;{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(R,{\mathbb{F}})$.

We claim that the correspondence in Proposition \[lem\_descrip\_diff\_opp\]$(ii)$ gives $$\mathcal{E} \,\,\cong\, \,{{\normalfont\text{Hom}}}_{{\mathbb{F}}}\bigl( {\mathbb{F}}[\mathbf{y}] / I, {\mathbb{F}}\bigr)
    \,\, \,\hookrightarrow\, \,\,{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,{\mathbb{F}}).$$ The isomorphism (\[eq\_isom\_inv\_sys\_char\_zero\]) implies that $V\cong V^\prime = \left(0:_{{\mathbb{F}}\left[\mathbf{y^{-1}}\right]} I\right)$. Since $I \supseteq {\mathcal{M}}^m$, it follows that $V^\prime \subseteq \left(0:_{{\mathbb{F}}\left[\mathbf{y^{-1}}\right]} {\mathcal{M}}^m\right)$. For each $0 \le j < m$, there is a perfect pairing $$\label{eq_perf_pairing}
        {\left[\frac{{\mathbb{F}}[\mathbf{y}]}{{\mathcal{M}}^m}\right]}_j \;\otimes_{\mathbb{F}}\; {\left[\left(0:_{{\mathbb{F}}\left[\mathbf{y^{-1}}\right]} {\mathcal{M}}^m\right)\right]}_{-j} \;\rightarrow \;{\mathbb{F}}, \quad \mathbf{y}^\alpha \otimes_{\mathbb{F}}\frac{1}{\mathbf{y}^\beta} \,\mapsto\, \mathbf{y}^\alpha\cdot\frac{1}{\mathbf{y}^\beta} = \begin{cases}
        1 \;\;\text{ if } \alpha = \beta\\
        0 \;\;\text{ otherwise},
        \end{cases}$$ where $\vert\alpha\rvert=\vert\beta\rvert=j$, induced by the usual multiplication. Hence, we get the isomorphisms $$\label{eq_isom_inv_sys_diff_ops}
        \left(0:_{{\mathbb{F}}\left[\mathbf{y^{-1}}\right]} {\mathcal{M}}^m\right) \;\cong\; {{\normalfont\text{Hom}}}_{\mathbb{F}}\left(\frac{{\mathbb{F}}[\mathbf{y}]}{{\mathcal{M}}^m},{\mathbb{F}}\right) \;\cong\; {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,{\mathbb{F}}) .$$ The second isomorphism follows from Proposition \[lem\_descrip\_diff\_opp\]$(i)$. The Hom-tensor adjunction yields $$\label{eq_isom_V_prime_dual_quot_I}
\!\!    \!\! V^\prime = \left(0 :_{\left(0:_{{\mathbb{F}}\left[\mathbf{y^{-1}}\right]} {\mathcal{M}}^m\right)} I\right)
        \cong\;{{\normalfont\text{Hom}}}_{{\mathbb{F}}[[\mathbf{y}]]}\left(\frac{{\mathbb{F}}[\mathbf{y}]}{I},
        {{\normalfont\text{Hom}}}_{\mathbb{F}}\left( \frac{{\mathbb{F}}[\mathbf{y}]}{{\mathcal{M}}^m},{\mathbb{F}}\right) \right) 
    \cong\; {{\normalfont\text{Hom}}}_{{\mathbb{F}}}\left(\frac{{\mathbb{F}}[\mathbf{y}]}{I}, {\mathbb{F}}\right).$$ The isomorphism $V^\prime \cong {{\normalfont\text{Hom}}}_{\mathbb{F}}\left(\frac{{\mathbb{F}}[\mathbf{y}]}{I},{\mathbb{F}}\right)$ also follows from the duality in [@EISEN_COMM Proposition 21.4].

By the isomorphism (\[eq\_isom\_inv\_sys\_char\_zero\]) and the map $\omega$ in (\[eq\_map\_omega\]), $\mathcal{E}$ can be obtained from $V^\prime$ via the map $$V^\prime \xrightarrow{\cong} \mathcal{E}, \quad
    \frac{1}{\mathbf{y}^\alpha} \mapsto \frac{1}{\alpha!}\partial_{\mathbf{x}}^\alpha.$$ On the other hand, by (\[eq\_diff\_opp\_z\_alpha\]), (\[eq\_perf\_pairing\]) and (\[eq\_isom\_inv\_sys\_diff\_ops\]), the dual monomial ${(\mathbf{y}^\alpha)}^* \in {{\normalfont\text{Hom}}}_{\mathbb{F}}\left(\frac{{\mathbb{F}}[y_1,\ldots,y_n]}{{\mathcal{M}}^m},{\mathbb{F}}\right)$ is identified with the inverted monomial $\frac{1}{\mathbf{y}^\alpha} \in {\mathbb{F}}[\mathbf{y^{-1}}]$ and with the differential operator $\overline{D_\mathbf{x}^\alpha} = \frac{1}{\alpha!}\overline{\partial_{\mathbf{x}}^\alpha}\in {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,{\mathbb{F}})$. Therefore, the isomorphisms in (\[eq\_isom\_V\_prime\_dual\_quot\_I\]) imply that $\mathcal{E}$ is indeed determined by $I$ via the correspondence in Proposition \[lem\_descrip\_diff\_opp\]$(ii)$.

After this identification, Proposition \[lem\_descrip\_diff\_opp\]$(iii)$ and Theorem \[thm:param\_primary\] imply that $${\rm Sol}(\mathcal{E})\, =\, \gamma^{-1}(I) \,=\,Q.$$ This completes the proof of Theorem \[thm\_noeth\_ops\_zero\_dim\], and we obtain Theorem \[thm:main\] for $P$ maximal.

The general case
----------------

In this subsection, we complete the proof of Theorem \[thm:main\]. As before, $R={\mathbb{K}}[x_1,\ldots,x_n]$, ${\rm char}({\mathbb{K}}) = 0$, and $P \subset R$ is a prime ideal of height $c$. We use the notation from Section \[sec4\], where $S={\mathbb{K}}(x_{c+1},\ldots,x_n)[x_1,\ldots,x_c]$ and ${\mathfrak{p}}= PS$. By Noether normalization, ${\mathbb{K}}[x_{c+1},\ldots,x_n] \hookrightarrow R/P$ is an integral extension. The ideal ${\mathfrak{p}}\subset S $ is maximal and ${\mathbb{F}}= S/{\mathfrak{p}}$. The following remarks will allow us to derive Theorem \[thm:main\] from Theorems \[thm:param\_primary\], \[thm:Macaulay\_dual\] and \[thm\_noeth\_ops\_zero\_dim\].

\[rem\_clear\_fractions\_diff\_ops\] By Lemma \[lem\_diff\_ops\_R/J\], any operator $A^\prime\in {{\normalfont\text{Diff}}}_{S/{\mathbb{K}}(x_{c+1},\ldots,x_n)}^{m-1}(S,S/{\mathfrak{p}})$ can be written as $$A^\prime \,\,\,= \sum_{\substack{\beta \in {\mathbb{N}}^c\\ \lvert \beta \rvert \le m-1}}
     \overline{h_\beta}\,\, \overline{\partial_{x_1}^{\beta_1}\cdots\partial_{x_c}^{\beta_c}}  
     \quad \text{ for some }\,\, h_\beta \in S.$$ We choose $h \in {\mathbb{K}}[x_{c+1},\ldots,x_n]$ such that $h\cdot h_\beta \in R$ for all $\beta$. Hence, we can consider $$A \,\,\, = \sum_{\substack{\beta \in {\mathbb{N}}^c\\ \lvert \beta \rvert \le m-1}} \overline{h\cdot h_\beta}\, \overline{\partial_{x_1}^{\beta_1}\cdots\partial_{x_c}^{\beta_c}}  \;\in \; {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R/P).$$ This differential operator satisfies $\,{\rm Sol}(A)={\rm Sol}(A^\prime) \cap R$.

\[rem\_lift\_diff\_ops\_sol\] Let $A^\prime = \sum_{\lvert \alpha \rvert \le m-1} \overline{r_\alpha} \overline{\partial_{\mathbf{x}}^\alpha} \,\in\, {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R/P)$ be a differential operator. By Lemma \[lem\_diff\_ops\_R/J\], we can lift this to $A = \sum_{\lvert \alpha \rvert \le m-1} 
r_\alpha \partial_\mathbf{x}^\alpha \,\in\, {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R)$. Then, it follows that $${\rm Sol}(A^\prime)=\lbrace f \in R : A \bullet f \in P \rbrace .$$

The next remark describes the Weyl-Noether module in terms of differential operators.

\[rem\_isom\_restrict\_Weyl\_mod\] We have the following isomorphisms $$\begin{aligned}
        {\mathbb{F}}\otimes_R D_{n,c} \,\,=\,\,
         {\mathbb{F}}\otimes_R R\langle\partial_{x_1},\ldots,\partial_{x_c} \rangle \,&
\,\cong\,\, {\mathbb{F}}\otimes_S \left( S \otimes_R R\langle \partial_{x_1},\ldots,\partial_{x_c}\rangle\right)\\
        & \,\cong \,\, {\mathbb{F}}\otimes_S  S\langle \partial_{x_1},\ldots,\partial_{x_c}\rangle\\
        & \,\cong\,\, {{\normalfont\text{Diff}}}_{S/{\mathbb{K}}(x_{c+1},\ldots,x_n)}\left(S,{\mathbb{F}}\right).
    \end{aligned}$$ The last isomorphism follows from (\[eq\_isom\_relWeyl\_diff\]) by applying this to the polynomial ring $S={\mathbb{K}}(x_{c+1},\ldots,x_n)[x_1,\ldots,x_c]$ and the maximal ideal ${\mathfrak{p}}= PS$ in $S$.

The correspondences between parts (a), (b) and (c) have already been established in Theorems \[thm:param\_primary\] and \[thm:Macaulay\_dual\]. Using Remark \[rem\_isom\_restrict\_Weyl\_mod\], we identify the Weyl-Noether module $\,{\mathbb{F}}\otimes_R D_{n,c} \,$ with $\, {{\normalfont\text{Diff}}}_{S/{\mathbb{K}}(x_{c+1},\ldots,x_n)}\left(S,{\mathbb{F}}\right)$. As in (\[eq\_map\_omega\]), we consider the map $$\begin{aligned}
        \label{eq_map_omega_S}
        \begin{split}
        \omega_S \,:\, {\mathbb{F}}[z_1,\ldots,z_c] \;&\rightarrow\; {\mathbb{F}}\,\otimes_R \,  D_{n,c}\;\cong\;{{\normalfont\text{Diff}}}_{S/{\mathbb{K}}(x_{c+1},\ldots,x_n)}(S,{\mathbb{F}})\\
        z_1^{\alpha_1}\cdots z_c^{\alpha_c} \;&\mapsto\; \partial_{x_1}^{\alpha_1}\cdots\partial_{x_c}^{\alpha_c},
        \end{split}
    \end{aligned}$$ but now applied to the polynomial ring $S={\mathbb{K}}(x_{c+1},\ldots,x_n)[x_1,\ldots,x_c]$ and its maximal ideal ${\mathfrak{p}}\subset S$. This map $\omega_S$ yields the correspondence between parts (c) and (d), that is, between $m$-dimensional ${\mathbb{F}}$-vector subspaces of ${\mathbb{F}}[z_1,\ldots,z_c]$ that are closed under differentiation and $m$-dimensional ${\mathbb{F}}$-vector subspaces of ${\mathbb{F}}\otimes_R D_{n,c}$ that are $R$-bi-modules under the action (\[eq\_deriv\_z\_bracket\_partial\]).

It remains to show that a basis of an ${\mathbb{F}}$-vector subspace in part (d) can be lifted to a set of Noetherian operators for the $P$-primary ideal in part (a). For that, let $Q$ be a $P$-primary ideal with multiplicity $m$ over $P$, and set $I=\gamma(Q)$, $V = I^\perp$ and $\mathcal{E} = \omega_S(V)$, by using Theorem \[thm:param\_primary\], Theorem \[thm:Macaulay\_dual\] and (\[eq\_map\_omega\_S\]), respectively. Then, Theorem \[thm\_noeth\_ops\_zero\_dim\] implies that, for any basis $A_1^{\prime\prime},\ldots,A_m^{\prime\prime}$ of the ${\mathbb{F}}$-vector subspace $\mathcal{E} \subset {{\normalfont\text{Diff}}}_{S/{\mathbb{K}}(x_{c+1},\ldots,x_n)}(S,{\mathbb{F}})$, we get the equality $
    QS = {\rm Sol}(A_1^{\prime\prime},\ldots,A_m^{\prime\prime}).
    $ From Remark \[rem\_clear\_fractions\_diff\_ops\], we can choose differential operators $$A_i^\prime \,\,=\,\, \sum_{\alpha \in {\mathbb{N}}^c} \overline{r_{i,\alpha}} \,\,
     \overline{\partial_{x_1}^{\alpha_1}\cdots\partial_{x_c}^{\alpha_c}} 
     \,\in\, {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(R,R/P), \quad \text{ where } 1\le i \le m \text{ and } r_{i,\alpha} \in R,$$ such that $Q={\rm Sol}(A_1^\prime,\ldots,A_m^\prime)$. Finally, by Remark \[rem\_lift\_diff\_ops\_sol\], the lifted differential operators $$A_i \,\,=\,\, \sum_{\alpha \in {\mathbb{N}}^c} r_{i,\alpha} \partial_{x_1}^{\alpha_1}\cdots\partial_{x_c}^{\alpha_c} \,\in\, D_{n,c}
    $$ are Noetherian operators for $Q$, which means that (\[eq:fromAtoQ\]) holds. This completes the proof.

Symbolic Powers and other Joins {#sec7}
===============================

The symbolic power of an ideal is a fundamental construction in commutative algebra. We here work in the polynomial ring $R={\mathbb{K}}[x_1,\ldots,x_n]$ over a field ${\mathbb{K}}$ of characteristic zero, with irrelevant maximal ideal $\mathfrak{m} = \langle x_1,\ldots,x_n \rangle$. The $r$-th [*symbolic power*]{} of an ideal $J$ in $R$ equals $$J^{(r)} \,\,\,\,:= \,\,\bigcap_{{\mathfrak{p}}\in {\rm Ass}(J)} \!\! J^r R_{\mathfrak{p}}\cap R.$$ Hence, if $P$ is a prime ideal in $R$ then $P^{(r)}$ is the $P$-primary component of the usual power $P^r$. If $\,{\rm codim}(P) = c\,$ then the primary ideal $P^{(r)}$ has multiplicity $m =  \binom{c+r-1}{c}$ over $P$, and in Theorem \[thm:param\_primary\] it is represented by the zero-dimensional ideal $\,I = \langle y_1,\ldots,y_c \rangle^r\, \subset \, {\mathbb{F}}[y_1,\ldots,y_c]$.

Our point of departure in this section is a formula due to Sullivant [@SULLIVANT_SYMB Proposition 2.8]: $$\label{eq:sulli1}
 J^{(r)} \,\, = \,\, J \star \mathfrak{m}^r .$$ Here, $J$ is any radical ideal in $R$, and $\star$ denotes the join of ideals. This is a reformulation of the [*Zariski-Nagata Theorem*]{} which expresses the symbolic power via differential equations: $$\label{eq:sulli2}
J^{(r)} \,\, = \,\, \,
\biggl\{ \,f \,\in\, R \,\,\,\bigg\vert \,\,\,
\frac{\partial^{i_1+i_2+\cdots+i_n} f}{
\partial x_1^{i_1}
\partial x_2^{i_2} \cdots
\partial x_n^{i_n} 
} \,\in\, J \quad \text{whenever}\, \,\,i_1+i_2 + \cdots + i_n < r \, \biggr\}.$$ The goal of this section is to generalize the equivalence between (\[eq:sulli1\]) and (\[eq:sulli2\]). We construct $P$-primary ideals by means of joins and we relate this to the results seen in earlier sections.

If $J$ and $K$ are ideals in $R$, then their *join* is the new ideal $$J \star K \,\,\,:=\,\,\, \Big( J(\mathbf{v}) \,+\, K(\mathbf{w}) \,+\,
     \langle x_i - v_i - w_i : 1 \le i \le n \rangle \Big) \,\,\cap \,\,R,$$ where $J(\mathbf{v})$ is the ideal $J$ with new variables $v_i$ substituted for $x_i$ and $K(\mathbf{w})$ is the ideal $K$ with $w_i$ substituted for $x_i$. The parenthesized ideal lives in a polynomial ring in $3n$ variables.

\[rem\_kernel\_map\_join\] Following Simis and Ulrich [@SIMIS_ULRICH_JOIN], the join $J \star K$ equals the kernel of the map $$\begin{aligned}
        R \;&\,\rightarrow \,\;  \frac{{\mathbb{K}}[v_1,\ldots,v_n,w_1,\ldots,w_n]}{J(\mathbf{v})+K(\mathbf{w})} \,\;\;\xleftrightarrow{\cong}\; R/J \otimes_{\mathbb{K}}R/K\\
        x_i \;&\, \mapsto \,\,\;\; \overline{v_i}+\overline{w_i}  \qquad\qquad\qquad\qquad\leftrightarrow \;\;\overline{x_i} \otimes_{\mathbb{K}}1 + 1 \otimes_{\mathbb{K}}\overline{x_i}.
    \end{aligned}$$ Hence, the quotient $\,R/\left(J\star K\right)\,$ can be identified with a subring of $\,R/J \otimes_{\mathbb{K}}R/K$.

The following result summarizes a few basic properties of the join construction.

\[prop\_properties\_join\] Let $J$ and $K$ be ideals in $R$. Then, the following statements hold:

(i) If $J = J_1 \cap J_2$, where $J_1,J_2 \subset R$ are ideals, then $J \star K = (J_1 \star K) \cap (J_2 \star K)$.

(ii) $\sqrt{J \star K} = \sqrt{J} \star \sqrt{K}$; in particular, $J \star K$ is radical when $J$ and $K$ are.

(iii) Suppose that ${\mathbb{K}}$ is algebraically closed. If $P_1$ and $P_2$ are prime ideals, then $P_1 \star P_2$ is a prime ideal. If $J$ and $K$ are primary ideals, then $J \star K$ is a primary ideal.

(iv) If $M$ is an $\mathfrak{m}$-primary ideal, then $P \star M$ is a $P$-primary ideal.

This is an adaptation of [@SIMIS_ULRICH_JOIN Proposition 1.2] for non-necessarily homogeneous ideals.

$(i)$ The join distributes over intersections by [@SULLIVANT_SYMB Lemma 2.6].

$(ii)$ The ring $R/\sqrt{J} \otimes_{\mathbb{K}}R/\sqrt{K}$ is reduced by [@GORTZ_WEDHORN Corollary 5.57]. As the kernel of the map $R/J\otimes_{\mathbb{K}}R/K \twoheadrightarrow R/\sqrt{J} \otimes_{\mathbb{K}}R/\sqrt{K}$ is nilpotent, the claim follows from Remark \[rem\_kernel\_map\_join\].

$(iii)$ Since ${\mathbb{K}}$ is algebraically closed, $R/P_1 \otimes_{\mathbb{K}}R/P_2$ is an integral domain [@GORTZ_WEDHORN Lemma 4.23]. By Remark \[rem\_kernel\_map\_join\], $R/(P_1 \star P_2)$ is a subring of this domain. Thus, $P_1 \star P_2$ is a prime ideal. Suppose ${\rm Ass}(R/J)=\{P_1\}$ and ${\rm Ass}(R/K)=\{P_2\}$. From [@MATSUMURA Theorem 23.2] we infer $$\label{eq_equality_ass_primes}
    {\rm Ass}(R/J \otimes_{\mathbb{K}}R/K)\,\, = \,\, {\rm Ass}(R/P_1 \otimes_{\mathbb{K}}R/P_2).$$ We already saw that $R/P_1 \otimes_{\mathbb{K}}R/P_2$ is an integral domain. Therefore, $R/J \otimes_{\mathbb{K}}R/K$ has only one associated prime, and hence its subring $R/(J \star K)$ has only one associated prime.

$(iv)$ The equality in (\[eq\_equality\_ass\_primes\]) is valid for any field. From this we get ${\rm Ass}(R/P \otimes_{\mathbb{K}}R/M)
     = {\rm Ass}(R/P \otimes_{\mathbb{K}}R/\mathfrak{m}) = \{ P \star \mathfrak{m} \} = \{P\}$. We hence conclude $\,{\rm Ass}(R / (P \star M))= \{P\}$.

In Proposition \[prop\_properties\_join\] (iii) we need the hypothesis that ${\mathbb{K}}$ is algebraically closed. If ${\mathbb{K}}= {\mathbb{R}}$ then $P_1 = \langle x_1^2+1, x_2 \rangle$ and $P_2 = \langle x_1 , x_2^2+1 \rangle $ are prime but their join is not primary: $$P_1 \star P_2 \,\, = \,\, \langle x_1^2+1, x_2^2+1 \rangle \,\, =\,\,
\langle x_1 - x_2 , x_2^2+1 \rangle \,\, \cap \,\, \langle x_1 + x_2 , x_2^2+1 \rangle .$$

In what follows we focus on the $P$-primary ideals $Q = P \star M$ in Proposition \[prop\_properties\_join\] (iv). These will be characterized by differential equations derived from the $\mathfrak{m}$-primary ideal $M$.

Let $M$ be an $\mathfrak{m}$-primary ideal. We shall encode $M$ by a system $\mathfrak{A}(M)$ of linear PDE with constant coefficients. This is computed by the performing following steps:

(i) Interpret $M$ as PDE by replacing the variables $x_i$ with $\partial_{z_i}$ for $i=1,\ldots,n$.

(ii) Compute the inverse system $M^\perp=\left\lbrace F\in {\mathbb{K}}[z_1,\ldots,z_n]: f\bullet F=0 \mbox{ for all }f\in M\right\rbrace$.

(iii) Let $\mathfrak{A}(M) \subset {\mathbb{K}}[\partial_{x_1},\ldots,\partial_{x_n}]$ be the image of $M^\perp$ under the map $\mathbf{z}^\alpha \mapsto \partial_{\mathbf{x}}^\alpha$.

We say that the ${\mathbb{K}}$-subspace $\mathfrak{A}(M)$ comprises the *differential operators associated to $M$*.

\[rem\_joins\_props\] (i) The space $\mathfrak{A}(M)$ is closed under taking brackets as in (\[eq\_deriv\_z\_bracket\_partial\]) and Theorem \[thm:Macaulay\_dual\].

\(ii) For any $r \ge 1$, we have $ \mathfrak{A}\left(\mathfrak{m}^r\right) = \bigoplus_{\lvert \alpha \rvert \le r-1} {\mathbb{K}}\, \partial_{\mathbf{x}}^\alpha $. Thus, $\mathfrak{A}(\mathfrak{m}^r)$ comprises the differential operators used in the Zariski-Nagata formula for symbolic powers; see (\[eq:sulli2\]) and [@EISEN_COMM §3.9].

The following result is a generalization of the classical Zariski-Nagata Theorem, to ideals obtained with the join construction. Of main interest is the situation when $J=P$ is prime.

\[thm:ourZN\] Let $J$ be any ideal in $R = {\mathbb{K}}[x_1,\ldots,x_n]$ and let $M$ be an $\mathfrak{m}$-primary ideal.

(i) The join of $J$ and $M$ equals $J \star M \;=\;  \big\lbrace f \in R : A \bullet f \in J \;
      \text{ for all }\; A \in \mathfrak{A}(M) \big\rbrace$.

(ii) If $J$ is radical and $r \in {\mathbb{N}}$ then $\,
            J^{(r)} \,=\, J \star \mathfrak{m}^r \,=\, 
            \big\lbrace f \in R : \partial_{\mathbf{x}}^\alpha \bullet f \in J \;
             \text{ for all }\; \lvert \alpha \rvert \le r-1 \big\rbrace $.

Let $n=4,c=2$, fix the prime ideal $P$ in (\[eq:twistedcubic1\]), and consider the $\mathfrak{m}$-primary ideal $ M = \langle x_1^2,x_2^2,x_3^2,x_4^2 \rangle$. The join $Q = P \star M$ is a $P$-primary ideal of multiplicity $m=11$. It is minimally generated by eight octics such as $\,x_1^8-4 x_1^6 x_2 x_3+6 x_1^4 x_2^2 x_3^2-4x_1^2 x_2^3 x_3^3+x_2^4 x_3^4$. The differential equations from $\mathfrak{A}(M)$ are simply the squarefree partial derivatives, so that $$\label{eq:repQ1}
 Q \,\,\,= \,\,\,\biggl\{ \,f \,\in\, R \,\,\,\bigg\vert \,\,\,
\frac{\partial^{i_1+i_2+i_3+i_4} f}{
\partial x_1^{i_1}
\partial x_2^{i_2}
\partial x_3^{i_3}
\partial x_4^{i_4} 
} \,\in\, P \quad \text{whenever}\, \,\,i_1,i_2,i_3,i_4 \in \{0,1\}\, \biggr\}.$$ This should be compared to the representation by Noetherian operators found in Algorithm \[alg:forward\]. In Step 1, we obtain the ideal $I = \langle y_1^4, u_2  y_1^3 y_2 - u_3  y_1 y_2^3, 3 u_1  y_1^2 y_2^2 - 5 u_3  y_1 y_2^3, y_2^4 \rangle$. The inverse system $I^\perp$ in Step 2 is the $11$-dimensional subspace of ${\mathbb{F}}[y_1,y_2]$ spanned by $$B({\bf u},{\bf z}) \,\,\,= \,\,\, 2 u_1 u_3 \,z_1^3 z_2\, +\, 5 u_2 u_3 \,z_1^2 z_2^2 
\,+\, 2 u_1 u_2 \,z_1 z_2^3$$ together with all ten monomials $z_1^{j_1} z_2^{j_2} $ of degree $j_1+j_2\leq 3$. From Steps 3 and 4 we obtain $$A({\bf x},\partial_{\bf x}) \,\, = \,\,
   2 x_1 x_3 \partial_{x_1}^3 \partial_{x_2 }
+ 5 x_2 x_3   \partial_{x_1}^2 \partial_{x_2}^2 
+ 2 x_1 x_2 \partial_{x_1} \partial_{x_2}^3  ,$$ and this gives rise to the following alternative representation of $Q$ by differential equations: $$\label{eq:repQ2}
 Q \,\,\,= \,\,\,\biggl\{ \,f \,\in\, R \,\,\,\bigg\vert \,\,\,
A \bullet f \in P \,\,\, \,{\rm and} \,\,\,
\frac{\partial^{j_1+j_2} f}{
\partial x_1^{j_1}
\partial x_2^{j_2}
} \,\in\, P \quad \text{whenever}\, \,j_1 + j_2 \leq 3
\, \biggr\}.$$ The two representations (\[eq:repQ1\]) and (\[eq:repQ2\]) differ in two fundamental ways. The operators in (\[eq:repQ1\]) have constant coefficients but differentiation involves all four variables. In (\[eq:repQ2\]) we are using an operator from $D_{4,2}$ with polynomial coefficients but we differentiate only two variables.

The next example shows that not every primary ideal arises from the join construction.

\[exam:Palamodov\] Let $n = 3$ and $c = 2$, and consider the primary ideal $Q = \langle x_1^2, x_2^2, x_1 - x_2x_3 \rangle$ with $P = \sqrt{Q}=\langle x_1, x_2 \rangle$. From [@BJORK Proposition 4.8 and Example 4.9, page 352] we know that $Q$ cannot be described by differential operators with constant coefficients only. Theorem \[thm:ourZN\] (i) implies that $Q$ does not arise from the join construction, i.e. we cannot find an $\mathfrak{m}$-primary ideal $M$ such that $Q = P \star M$. On the other hand, Algorithm \[alg:forward\] applied to $Q$ gives the two Noetherian operators $A_1 = 1, A_2 = x_3\partial_{x_1} + \partial_{x_2}$.

$(i)$ We use the notation and results from Section \[sec5\]. We begin by fixing an integer $m$ such that $\mathfrak{m}^m \subseteq M$. In (\[eq\_isoms\_Diff\_R/J\]) we obtained the explicit isomorphism $$\label{eq_join_isom_diff_ops}
            {{\normalfont\text{Hom}}}_{R/J}\left(R/J \otimes_R P_{R/{\mathbb{K}}}^{m-1}, R/J\right) \xrightarrow{\cong} {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R/J), \quad \psi \mapsto \psi \circ h_{m-1} \circ d^{m-1}$$ where $h_{m-1}$ is the canonical map $ P_{R/{\mathbb{K}}}^{m-1} \rightarrow R/J \otimes_R P_{R/{\mathbb{K}}}^{m-1}$ and $d^{m-1}$ is the map in (\[eq\_univ\_diff\]). Setting $\,T=R \otimes_{\mathbb{K}}R ={\mathbb{K}}[x_1, \ldots, x_n, y_1, \ldots, y_n]\,$ as in Section \[sec5\], we have the following isomorphisms: $$\label{eq_isom_tensor_prods_join}
    R/J \otimes_R P_{R/{\mathbb{K}}}^{m-1}  \;\cong\; \frac{T}{J(\mathbf{x}) \,+
    \, \mathfrak{m}^m(\mathbf{y})} \;\cong\; R/J \otimes_{\mathbb{K}}R/\mathfrak{m}^m.$$ Recall that this ${\mathbb{K}}$-vector space is considered as an $R$-module via the left factor $R/J \otimes_{\mathbb{K}}1$.

Using (\[eq\_join\_isom\_diff\_ops\]) and (\[eq\_isom\_tensor\_prods\_join\]), the surjection $\,R/J \otimes_{\mathbb{K}}R/\mathfrak{m}^m \twoheadrightarrow R/J \otimes_{\mathbb{K}}R/M\,$ induces the inclusion $$\label{eq_inclusion_join_diffs}
            {{\normalfont\text{Hom}}}_{R/J}\left(R/J \otimes_{\mathbb{K}}R/M, R/J\right) 
            \,\,\hookrightarrow\,\, {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R/J).$$ Since $ R/J \otimes_{\mathbb{K}}R/M$ is a finitely generated free $R/J$-module, we have $$\big\lbrace w \in  R/J \otimes_{\mathbb{K}}R/M : \psi(w)=0 \;\text{ for all }\; \psi \in {{\normalfont\text{Hom}}}_{R/J}\left(R/J \otimes_{\mathbb{K}}R/M, R/J\right) \big\rbrace \;=\;  \{0\}.$$ Let $\,\mathcal{E} \subseteq {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R/J)\,$ denote the image of (\[eq\_inclusion\_join\_diffs\]). So, the isomorphism (\[eq\_join\_isom\_diff\_ops\]) implies $${\rm Sol}(\mathcal{E}) = {\rm Ker}\left(\overline{d^{m-1}}\right), \;\text{ where }\; \overline{d^{m-1}} : R \rightarrow R/J \otimes_{\mathbb{K}}R/M, \;\; x_i \mapsto \overline{x_i} \otimes_{\mathbb{K}}1 + 1 \otimes_{\mathbb{K}}\overline{x_i}.$$ Therefore, Remark \[rem\_kernel\_map\_join\] yields that ${\rm Sol}(\mathcal{E})=J \star M$.

By [@MATSUMURA Theorem 7.11], the inclusion (\[eq\_inclusion\_join\_diffs\]) can be written equivalently as $$\begin{aligned}
    {{\normalfont\text{Hom}}}_{R/J}\left(R/J \otimes_{\mathbb{K}}R/M, R/J\right)
            \,\, \cong \,\, R/J &\otimes_{\mathbb{K}}{{\normalfont\text{Hom}}}_{\mathbb{K}}(R/M,{\mathbb{K}})\\ &
    \hookrightarrow\; R/J \otimes_{\mathbb{K}}{{\normalfont\text{Hom}}}_{\mathbb{K}}(R/\mathfrak{m}^m,{\mathbb{K}}) \,\, \cong \,\, {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R/J).\end{aligned}$$ The Hom-tensor adjunction and the perfect pairing in (\[eq\_perf\_pairing\]) give the following isomorphisms: $${{\normalfont\text{Hom}}}_{\mathbb{K}}\left(R/M,{\mathbb{K}}\right) \,\,\cong\,\, {{\normalfont\text{Hom}}}_R\left(R/M, {{\normalfont\text{Hom}}}_{{\mathbb{K}}}\left(R/\mathfrak{m}^m,{\mathbb{K}}\right)\right) \,\,\cong\,\, \left(0 :_{{\mathbb{K}}[\mathbf{x^{-1}}]} M\right).$$ Then, by arguments almost verbatim to those used in the proof of Theorem \[thm\_noeth\_ops\_zero\_dim\], we find that $\,\mathcal{E} \subseteq {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R, R/J)\,$ is a finitely generated free $R/J$-module, and it is generated by $\,\bigl\{ \,\overline{A} \,:\,  A \in \mathfrak{A}(M) \subset {\mathbb{K}}[\partial_{\mathbf{x}}] \cap {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R)\bigr\} \subset {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R/J)$. Summing up, we conclude $$J \star M \,\,=\,\, {\rm Sol}(\mathcal{E}) \,\,=\,\,
   \big\lbrace f \in R : A \bullet f \in J \;\text{ for all }\; A \in \mathfrak{A}(M) \big\rbrace.$$

$(ii)$ Since $J$ is radical, $J=P_1 \cap \cdots \cap P_k$ for some prime ideals $P_j \subset R$, and so we have $J^{(r)}=P_1^{(r)} \cap \cdots \cap P_k^{(r)}$. Proposition \[prop\_properties\_join\]$(i)$ implies $J \star \mathfrak{m}^r = \left(P_1 \star \mathfrak{m}^r\right) \cap \cdots \cap \left(P_k \star \mathfrak{m}^r \right)$. Therefore, to finish the proof, it suffices to consider the case where $J=P$ is a prime ideal. The Zariski-Nagata Theorem implies $\,
 P^{(r)} = \big\lbrace f \in R : \partial_{\mathbf{x}}^\alpha \bullet f \in J \; \text{ for all }\; \lvert \alpha \rvert \le r-1 \big\rbrace $. The conclusion now follows from part $(i)$ applied to $M=\mathfrak{m}^r$. This establishes Theorem \[thm:ourZN\].

Decomposition and Fusion in a Numerical Future {#sec8}
==============================================

This closing section takes the perspective of applied and computational mathematics. We consider a system of polynomial equations over the complex numbers ${\mathbb{C} }$, viewed as an ideal $I$ in the polynomial ring $R = {\mathbb{C} }[x_1,\ldots,x_n]$. This ideal has a minimal primary decomposition $$\label{eq:primdeco}
 I \,\,\, = \,\,\, Q_1 \,\cap \, Q_2 \,\cap \, \cdots \,\cap \, Q_s .$$ Each associated prime $P_i = \sqrt{Q_i}$ defines an irreducible variety $X_i = V(P_i)$ in ${\mathbb{C} }^n$. Solving the equations means identifying the varieties $X_i$ corresponding to the associated primes $P_i$.

Computing the primary decomposition (\[eq:primdeco\]) from generators of $I$ thus refines the problem of solving polynomial systems. Algorithms for this task are a well-developed subject in computer algebra [@DECKER]. However, most studies focus on the irreducible components $X_i$ and the associated primes $P_i$, and they pay less attention to the primary ideals $Q_i$ themselves.

The past decade has seen significant advances in numerical algebraic geometry [@BATES], and this has led to the design of numerical techniques for primary decomposition [@KRONE; @LEYKIN]. A paramount ingredient is the identification of all minimal primes $P_i$ from the generators of $I$. Algorithms and implementations for this are now well-established; see e.g. [@BATES Chapter 8]. In the output, each irreducible variety $X_i$ is represented by a finite [*witness set*]{} of the form $X_i \cap L_i$, where $L_i$ is a general affine-linear subspace of dimension $c_i = {\rm codim}(X_i)$ in ${\mathbb{C} }^n$.

Numerical identification of embedded primes $P_i$ is more subtle. This topic was pioneered by Krone and Leykin [@KRONE; @LEYKIN] who proposed algorithms based on a technique known as [*inflation*]{}. However, the concluding paragraph in [@KRONE] indicates that more work is needed. Furthermore, their articles do not address the description of the primary ideals $Q_i$ in (\[eq:primdeco\]).

The following definitions pave the way for future numerical algorithms. By Theorem \[thm:main\], each primary ideal $Q_i$ is encoded by a pair $(X_i,\mathfrak{A}_i$) where $\mathfrak{A}_i$ is an $m_i$-dimensional ${\mathbb{F}}_i$-vector subspace of ${\mathbb{F}}_i \otimes_R D_{n,c_i}$, where $X_i = V(P_i) = V(Q_i)$ and ${\mathbb{F}}_i $ denotes the field of fractions of $R/P_i$. The numerical representation of the prime ideal $P_i$ or the associated function field ${\mathbb{F}}_i$ is the same as that of $X_i$, namely it is simply a witness set as in [@BATES Chapter 8]. The space $\mathfrak{A}_i$ provides a set of Noetherian operators $A_{ij}({\bf x},\partial_{\bf x})$ for $Q_i$, where $j=1,2,\ldots,m_i$.

We propose to use (\[eq:fromAtoQ\]) as the numerical encoding of primary ideals in future algorithms: $$\label{eq:fromAtoQnum}
 Q_i \,\, = \,\, \{ \, f \in R \,: \, A_{ij} \bullet f \,\,\text{vanishes on} \,\, X_i \,
\,\text{for all} \,\, j \,\}.$$ Here $A_{ij}$ is an element in the relative Weyl algebra $D_{n,c_i}$ and its coefficients are given in floating point arithmetic. Likewise, the vanishing condition in (\[eq:fromAtoQnum\]) is meant to be inexact.

\[def:numprimdec\] Given an ideal $I$ in $R = {\mathbb{C} }[x_1,\ldots,x_n]$, we define a [*numerical primary decomposition*]{} of $I$ to be a list $(X_1,\mathfrak{A}_1), \ldots, (X_s,\mathfrak{A}_s)$ of representations of primary ideals, where the $X_i$ are precisely the irreducible varieties that are associated to $I$, and we have $$\label{eq:fusionI}
 I \,\, = \,\, \{ \, f \in R \,: \, A \bullet f \,\,\,\text{vanishes on}\, \, X_i\,
    \,\,\text{for all $A \in \mathfrak{A}_i $ and all $i=1,\ldots,s$} \,\}.$$ By an abuse of notation, here each $\mathfrak{A}_i$ is also identified with an appropriate finite subset of Noetherian operators for $Q_i$. If $X_i$ is a geometric component then this subset is simply obtained from a basis of the relevant $R$-bi-module in part (d) of Theorem \[thm:main\], and its cardinality is the multiplicity $m_i$ of the primary ideal $Q_i$. However, if $X_i$ is an embedded component, say $X_i \subset X_j$, then we may use a subset of cardinality strictly less than $m_i$.

Let $n=2$ and $I = \langle x_1^3, x_1^2 x_2^2\rangle\, = \, \langle x_1^2 \rangle \,\cap \,
    \langle x_1^3, x_2^2 \rangle$. A numerical primary decomposition consists of $(X_1,\mathfrak{A}_1)$ and $(X_2,\mathfrak{A}_2)$, where $X_1 $ is the $x_2$-axis with $\mathfrak{A}_1 = \{ 1, \partial_1 \}$, and $X_2 = \{(0,0)\}$ with $\mathfrak{A}_2 = \{ 
    \partial_1^2 \partial_2, \partial_1^2 , \partial_1 \partial_2, \partial_2 \}$. Note that $|\mathfrak{A}_2|=4 < 6 = m_2 = {\rm mult}(\langle x_1^3,x_2^2\rangle) $.

The computation of a numerical primary decomposition should be carried out by combining existing methods for numerical irreducible decomposition [@BATES; @KRONE; @LEYKIN] with an appropriate adaptation of Algorithm \[alg:forward\]. For each associated irreducible variety $X_i$ one must identify the inverse system in Step 2 using linear algebra over the function field ${\mathbb{F}}_i$ of the component $X_i$. Linear algebra over ${\mathbb{F}}_i$ is to be carried out not from equations but from the witness set alone.

One task that arises naturally in this setting is the converse to primary decomposition. This process, which we propose to call [*primary fusion*]{}, amounts to combining a finite collection of primary ideals by their intersection. Let $Q_1$ and $Q_2$ be primary ideals in $R$, encoded by pairs $(X_1,\mathfrak{A}_1)$ and $(X_2,\mathfrak{A}_2)$ as above. Here $\mathfrak{A}_i$ is an $R$-bi-submodule of ${\mathbb{F}}_i \otimes_R D_{n,c_i}$. The first case to consider is when the underlying varieties agree, so $X_1 = X_2$ with $c = c_1 = c_2$.

If $Q_1$ and $Q_2$ are $P$-primary ideals, then $Q_1 \cap Q_2$ is also $P$-primary. Its bi-module of Noetherian operators in ${\mathbb{F}}\otimes_R D_{n,c}$ is $\,\mathfrak{A}_1 +  \mathfrak{A}_2$. This is the primary fusion.

Next consider the situation when $P_1 = \sqrt{Q_1}$ and $P_2 = \sqrt{Q_2}$ are distinct. Suppose first that there is no containment between the varieties $X_1 $ and $X_2$. We certify this from their witness sets. In this case, the primary fusion of $(X_1,\mathfrak{A}_1)$ and $(X_2,\mathfrak{A}_2)$ is the union of their representations by Noetherian operators, that is, the primary fusion is (\[eq:fusionI\]) with $s=2$.

The most interesting case arises when $X_1 \subset X_2$. The codimensions satisfy $c_1 > c_2$ and coordinates are chosen so that the Noether normalizations are compatible. Note that $\mathfrak{A}_1 \subset {\mathbb{F}}_1 \otimes_R D_{n,c_1} $ and $\mathfrak{A}_2 \subset {\mathbb{F}}_2 \otimes_R D_{n,c_2} $. We wish to replace the ${\mathbb{F}}_1$-vector space $\mathfrak{A}_1$ by a proper subspace in order to turn (\[eq:fusionI\]) into a minimal representation for $Q_1 \cap Q_2$. It would be desirable to develop an algorithm for doing this in practice, not just for two components but for an arbitrary number $s$ of numerically represented primary ideals $(X_i,\mathfrak{A}_i)$.

\[prob:fusion\] Develop a practical numerical method for primary fusion in ${\mathbb{C} }[x_1,\ldots,x_n]$.

The numerical solution of partial differential equations is a vast area whose importance for the sciences and engineering can hardly be overestimated. In this paper we explored one special aspect, namely systems of homogeneous linear PDE on ${\mathbb{C} }^n$ with constant coefficients. Such PDE are polynomials in the operators $ \partial_{z_1},
\ldots, \partial_{z_n}$, and their solutions are functions $\psi(z_1,\ldots,z_n)$. We seek numerical algorithms for computing and manipulating these $\psi({\bf z})$ via their integral representations (\[eq:anysolution\]), promised to us by Ehrenpreis [@EHRENPREIS] and Palamodov [@PALAMODOV]. These should go well beyond the zero-dimensional case, studied by Gröbner in the 1930’s.

The given PDE form an ideal $I$ in the polynomial ring $R$. We view this input and the desired output in the spirit of numerical algebraic geometry [@BATES]. Exploiting the primary decomposition (\[eq:primdeco\]), our task is to numerically compute the objects of Theorem \[thm:Palamodov\_Ehrenpreis\] for each primary ideal $Q_i$. The varieties $X_i$ are given by witness sets. These need to be enhanced by measures $\mu_{ij}$ for the integral representation (\[eq:anysolution\]). The key algebraic objects are the Noetherian multipliers $B_{ij}({\bf x},{\bf z})$. Their construction is described in Step 3 of Algorithm \[alg:forward\], but this must now be done in a numerical setting. Moreover, to combine solutions $\psi_1({\bf z})$ and $\psi_2({\bf z})$ whose supports are nested, say $X_1 \subset X_2$, we also need primary fusion (Problem \[prob:fusion\]).

The problem of solving linear PDE with constant coefficients was discussed in [@STURMFELS_SOLVING Chapter 10]. The author of [@STURMFELS_SOLVING] worked out several nice examples, like the one of page 144, but he was unable to go further, because he lacked the necessary tools from commutative algebra.

Overcoming that barrier is precisely the contribution of the present paper. We here develop the tools from commutative algebra that were needed to advance [@STURMFELS_SOLVING Chapter 10]. Theorem \[thm:main\] offers a new characterization of primary ideals and their differential equations. This leads to Algorithms \[alg:forward\] and \[alg:backward\], and these lay the foundation for future development of the Ehrenpreis-Palamodov Fundamental Principle within numerical algebraic geometry.

[10]{}

Daniel Bates, Jonathan Hauenstein, Andrew Sommese and Charles Wampler: [*Numerically Solving Polynomial Systems with Bertini*]{}, Software, Environments, Tools: Vol 25, SIAM, 2013.

Jan-Erik Björk: [*Rings of Differential Operators*]{}, volume 21 of [ *North-Holland Mathematical Library*]{}, North-Holland Publishing Co., Amsterdam-New York, 1979.

Joël Briançon: Description de ${\rm Hilb}^n {\mathbb{C} }[[x,y]]$, [*Inventiones mathematicae*]{} [**41**]{} (1977) 45 –90.

Markus Brodmann and Rodney Sharp: [*Local Cohomology*]{}, volume 136 of [*Cambridge Studies in Advanced Mathematics*]{}, Cambridge University Press, second edition, 2013.

Gregory Brumfiel: Differential operators and primary ideals, [*J. Algebra*]{} [**51**]{} (1978) 375–398.

Winfried Bruns and Jürgen Herzog: [*Cohen-[M]{}acaulay Rings*]{}, Cambridge Studies in Advanced Mathematics. Cambridge University Press, second edition, 1998.

Yairon Cid-Ruiz: Noetherian operators, primary submodules and symbolic powers, [arXiv:1909.07253]{}.

Alberto Damiano, Irene Sabadini and Daniele Struppa: Computational methods for the construction of a class of Noetherian operators, [*Experimental Mathematics*]{} [**16**]{} (2007) 41–53.

Wolfram Decker, Gert-Martin Greuel and Gerhard Pfister: Primary decomposition: algorithms and comparisons, in: Matzat B.H., Greuel GM., Hiss G. (eds) [*Algorithmic Algebra and Number Theory*]{}, 187–220, Springer, Berlin, Heidelberg

Leon Ehrenpreis: [*Fourier Analysis in Several Complex Variables*]{}, Pure and Applied Mathematics, Vol. XVII, Wiley-Interscience Publishers, New York-London-Sydney, 1970.

David Eisenbud: [*Commutative Algebra with a View Towards Algebraic Geometry*]{}, Graduate Texts in Mathematics, Vol 150, Springer-Verlag, 1995.

Ulrich Görtz and Torsten Wedhorn: [*Algebraic Geometry [I]{}. Schemes with Examples and Exercises*]{}, Advanced Lectures in Mathematics, Vieweg and Teubner, Wiesbaden, 2010.

Daniel Grayson and Michael Stillman: Macaulay2, a software system for research in algebraic geometry, available at <http://www.math.uiuc.edu/Macaulay2/>.

Wolfgang Gröbner: Über das Macaulaysche inverse System und dessen Bedeutung für die Theorie der linearen Differentialgleichungen mit konstanten Koeffizienten, [*Abh. Math. Semin. Univ. Hamburg*]{} [**12**]{} (1937) 127–132.

Wolfgang Gröbner: Über eine neue idealtheoretische [G]{}rundlegung der algebraischen [G]{}eometrie, [*Mathematische Annalen*]{} [**115**]{} (1938) 333–358.

Wolfgang Gröbner: Über die algebraischen Eigenschaften der Integrale von linearen Differentialgleichungen mit konstanten Koeffizienten, [*Monatshefte für Mathematik und Physik*]{} [**47**]{} (1939) 247–284.

Wolfgang Gröbner: La théorie des idéaux et la géométrie algébrique, in [*Deuxième [C]{}olloque de [G]{}éométrie [A]{}lgébrique, [L]{}iège, 1952*]{}, pages 129–144, Masson & Cie, Paris, 1952.

Wolfgang Gröbner: [*Algebraische [G]{}eometrie. [$2$]{}. [T]{}eil: [A]{}rithmetische [T]{}heorie der [P]{}olynomringe*]{}, Bibliographisches Institut, Mannheim-Vienna-Zurich, 1970.

Alexander [Grothendieck]{}. [É]{}léments de géométrie algébrique: [IV]{}. [É]{}tude locale des schémas et des morphismes de schémas, [*Publications Mathématiques de l’IHÉS*]{}, [**32**]{} (1967) 5–361.

Robert Heyneman and Moss Sweedler: Affine [H]{}opf algebras. [I]{}, [*J. Algebra*]{} [**13**]{} (1969) 192–241.

Lars Hörmander: [*An Introduction to Complex Analysis in Several Variables*]{}, volume 7 of [*North-Holland Mathematical Library*]{}, North-Holland, Amsterdam, third edition, 1990.

Anthony Iarrobino: Punctual [H]{}ilbert schemes, [*Bull. Amer. Math. Soc.*]{} [**78**]{} (1972) 819–823.

Robert Krone and Anton Leykin: Numerical algorithms for detecting embedded components, [*Journal of Symbolic Computation*]{} [**82**]{} (2017) 1–18.

Anton Leykin: Numerical primary decomposition, ISSAC ’08, Proceedings of the twenty-first [*International Symposium on Symbolic and Algebraic Computation*]{} (Linz 2008), 165–172.

John Little: The many lives of the twisted cubic, [*Amer. Math. Monthly*]{} [**126**]{} (2019) 579–592.

Hideyuki Matsumura: [*Commutative Ring Theory*]{}, Cambridge Studies in Advanced Mathematics, volume 8, Cambridge University Press, 1989.

Bernard Mourrain: Isolated points, duality and residues, [*J. Pure Appl. Algebra*]{} [**117/118**]{} (1997) 469–493, Algorithms for algebra (Eindhoven, 1996).

Ulrich Oberst: The construction of [N]{}oetherian operators, [*J. Algebra*]{} [**222**]{} (1999) 595–620.

Victor P. Palamodov: [*Linear Differential Qperators with Constant Coefficients*]{}, Grundlehren der mathematischen Wissenschaften, Band 168. Springer-Verlag, New York-Berlin, 1970.

Anna-Laura Sattelberger and Bernd Sturmfels: D-modules and holonomic functions, [arXiv:1910.01395]{}.

Francesco Severi: Les images géométriques des idéaux de polynômes, [*Comptes Rendus Acad. Sc. Paris*]{} [**232**]{} (1951) 2395–2396.

Aron Simis and Bernd Ulrich: On the ideal of an embedded join, [*J. Algebra*]{} [**226**]{} (2000) 1–14.

Bernd Sturmfels: [*Solving Systems of Polynomial Equations*]{}, volume 97 of [*CBMS Regional Conference Series in Mathematics*]{}, American Mathematical Society, Providence, RI, 2002.

Seth Sullivant: Combinatorial symbolic powers, [*J. Algebra*]{} [**319**]{} (2008) 115–142.

Yairon Cid-Ruiz, MPI-MiS Leipzig

Roser Homs, MPI-MiS Leipzig

Bernd Sturmfels, MPI-MiS Leipzig and UC Berkeley
