### Dual representations of polynomial modules with applications to partial differential equations

$$\begin{gather*} \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 g}{\partial x \partial y} = 0 \\ \frac{\partial^2 f }{\partial y \partial z } + \frac{\partial^2 g}{\partial z^2} = 0 \\ \frac{\partial^3 f }{\partial x^2 \partial z } + \frac{\partial^3 g}{\partial z \partial y \partial w} = 0 \end{gather*}$$

Marc Härkönen
PhD dissertation defense,
Apr 15 2022

# Example

$$I = (x_1^2, x_1 x_2) \subset R = \mathbb{Q}[x_1, x_2]$$

• Primary decomposition $I = (x_1) \cap (x_1^2, x_2)$.
• "Differential" primary decomposition $$I = \left\{ f \in R \colon f \text{ vanishes on the line x_1=0 and } \frac{\partial f}{\partial x_1} \text{ vanishes at the origin} \right\}$$

# Example

$$I = (x_1^2, x_1 x_2) \subset R = \mathbb{Q}[x_1, x_2]$$

• Primary decomposition $I = (x_1) \cap (x_1^2, x_2)$.
• "Differential" primary decomposition \begin{aligned} I = \{ f \in R \colon f \text{ vanishes on the line x_1=0 and }& \\ \frac{\partial f}{\partial x_1} \text{ vanishes at the origin} &\} \end{aligned}

$$\frac{\partial^2 \phi}{\partial z_1^2} = \frac{\partial^2 \phi}{\partial z_1 \partial z_2} = 0$$

Find $\phi(z_1,z_2) \colon \RR^2 \to \CC$ satisfying the above

# Example

$$I = (x_1^2, x_1 x_2) \subset R = \mathbb{Q}[x_1, x_2]$$

• Primary decomposition $I = (x_1) \cap (x_1^2, x_2)$.
• "Differential" primary decomposition \begin{aligned} I = \{ f \in R \colon f \text{ vanishes on the line x_1=0 and }& \\ \frac{\partial f}{\partial x_1} \text{ vanishes at the origin} &\} \end{aligned}

$$\frac{\partial^2 \phi}{\partial z_1^2} = \frac{\partial^2 \phi}{\partial z_1 \partial z_2} = 0$$

Find $\phi(z_1,z_2) \colon \RR^2 \to \CC$ satisfying the above

Solution: $\phi(z_1,z_2) = \psi(z_2) + c_1 z_1$,

$c_1 \in \CC,~~\psi(z_2) \colon \RR \to \CC$ differentiable.

# Example

$$I = (x_1^2, x_1 x_2) \subset R = \mathbb{Q}[x_1, x_2]$$

• Primary decomposition $I = (x_1) \cap (x_1^2, x_2)$.
• "Differential" primary decomposition \begin{aligned} I = \{ f \in R \colon f \text{ vanishes on the line x_1=0 and }& \\ \frac{\partial f}{\partial x_1} \text{ vanishes at the origin} &\} \end{aligned}

$$\frac{\partial^2 \phi}{\partial z_1^2} = \frac{\partial^2 \phi}{\partial z_1 \partial z_2} = 0$$

Find $\phi(z_1,z_2) \colon \RR^2 \to \CC$ satisfying the above

Solution: $\phi(z_1,z_2) = \psi(z_2) + c_1 z_1$,

$c_1 \in \CC,~~\psi(z_2) \colon \RR \to \CC$ differentiable.

Dualities at two levels

• ideals and their differential equations,
• differential equations and their solutions

# Gröbner duality

$$R = \KK[x_1,\dotsc,x_n], \qquad p \in \KK^n$$

\begin{aligned} \partial_i \colon R &\to \KK \\ f &\mapsto \partial_i \bullet f = \frac{\partial f}{\partial x_i}(p). \end{aligned}

$$W_{\KK} := \KK[\partial_1,\dotsc,\partial_n], \qquad \partial^\alpha \bullet f = \frac{\partial^{|\alpha|} f}{\partial x^\alpha}(p)$$

• A $\KK$-vector space
• A right $R$-module: $(Dg) \bullet f := D \bullet (gf)$

$$W_{\KK} := \KK[\partial_1,\dotsc,\partial_n], \qquad \partial^\alpha \bullet f = \frac{\partial^{|\alpha|} f}{\partial x^\alpha}(p)$$

• A $\KK$-vector space
• A right $R$-module: $(Dg) \bullet f := D \bullet (gf)$

### Definition

The dual space of $I \subseteq R$ at the point $p$ is $$I^\perp = \{ D \in W_\KK \colon D \bullet f = 0 \text{ for all } f \in I \}$$ For $\Lambda \subseteq \KK[\partial]$, we have a set of polynomial solutions $$\Lambda^\perp = \{f \in R \colon D \bullet f = 0 \text{ for all } D \in \Lambda \}$$

$$W_{\KK} := \KK[\partial_1,\dotsc,\partial_n], \qquad \partial^\alpha \bullet f = \frac{\partial^{|\alpha|} f}{\partial x^\alpha}(p)$$

• A $\KK$-vector space
• A right $R$-module: $(Dg) \bullet f := D \bullet (gf)$

### Definition

The dual space of $I \subseteq R$ at the point $p$ is $$I^\perp = \{ D \in W_\KK \colon D \bullet f = 0 \text{ for all } f \in I \}$$ For $\Lambda \subseteq \KK[\partial]$, we have a set of polynomial solutions $$\Lambda^\perp = \{f \in R \colon D \bullet f = 0 \text{ for all } D \in \Lambda \}$$

### Theorem (Gröbner)

Let $p \in \KK^n$, $\mm = (x_1 - p_1, \dotsc, x_n - p_n)$. There is a bijection between $\mm$-primary ideals in $R$ and finite dimensional right $R$-modules in $W_\KK$.

# Local dual spaces

Suppose $\mm \subset R$ is an arbitrary maximal ideal, $\kappa(\mm) = R/\mm$ the residue field.

\begin{aligned} \partial_i \colon R &\to \kappa(\mm) \\ f &\mapsto \partial_i \bullet f = \frac{\partial f}{\partial x_i} \mod{\mm} \end{aligned}

$$W_{\kappa(\mm)} := \kappa(\mm)[\partial_1,\dotsc,\partial_n], \qquad \partial^\alpha \bullet f = \frac{\partial^{|\alpha|} f}{\partial x^\alpha} \mod{\mm}$$

• A $\kappa(\mm)$-vector space
• A right $R$-module: $(Dg) \bullet f := D \bullet (gf)$

$$W_{\kappa(\mm)} := \kappa(\mm)[\partial_1,\dotsc,\partial_n], \qquad \partial^\alpha \bullet f = \frac{\partial^{|\alpha|} f}{\partial x^\alpha} \mod{\mm}$$

• A $\kappa(\mm)$-vector space
• A right $R$-module: $(Dg) \bullet f := D \bullet (gf)$

### Definition

The local dual space of $I \subseteq R$ at the $\mm$ is $$D_\mm[I] = \{ D \in W_{\kappa(\mm)} \colon D \bullet f = 0 \text{ for all } f \in I \}$$ For $\Lambda \subseteq W_{\kappa(\mm)}$, we have a set of polynomial solutions $$I_\mm[\Lambda] = \{f \in R \colon D \bullet f = 0 \text{ for all } D \in \Lambda \}$$ If $\Lambda \subseteq W_{\kappa(\pp)}$ is a $\kappa(\pp)$-vector space and a right $R$-module, we say that $\Lambda$ is a local dual space.

$$W_{\kappa(\mm)} := \kappa(\mm)[\partial_1,\dotsc,\partial_n], \qquad \partial^\alpha \bullet f = \frac{\partial^{|\alpha|} f}{\partial x^\alpha} \mod{\mm}$$

• A $\kappa(\mm)$-vector space
• A right $R$-module: $(Dg) \bullet f := D \bullet (gf)$

### Definition

The local dual space of $I \subseteq R$ at the $\mm$ is $$D_\mm[I] = \{ D \in W_{\kappa(\mm)} \colon D \bullet f = 0 \text{ for all } f \in I \}$$ For $\Lambda \subseteq W_{\kappa(\mm)}$, we have a set of polynomial solutions $$I_\mm[\Lambda] = \{f \in R \colon D \bullet f = 0 \text{ for all } D \in \Lambda \}$$ If $\Lambda \subseteq W_{\kappa(\pp)}$ is a $\kappa(\pp)$-vector space and a right $R$-module, we say that $\Lambda$ is a local dual space.

### Theorem

Let $\mm$ be a maximal ideal. We have bijections
• $\mm$-primary ideals in $R$ $\leftrightarrow$ finite dimensional local dual spaces in $W_{\kappa(\mm)}$
• $\mm$-closed ideals in $R$ $\leftrightarrow$ local dual spaces in $W_{\kappa(\mm)}$

$I \subseteq R$ is $\mm$-closed if $I_\mm \cap R = I$.

## Example

$R = \RR[x,y], \qquad \mm = (x-y, y^2+1)$ $$I = (2xy-y^2+1, x^2 + 1)$$ $$D_{\mm}[I] = \Span_{\kappa(\mm)}\{ 1, \partial_y \}$$
$\Lambda = \langle \partial_x^2 + \partial_x\partial_y,\,\partial_y^2 \rangle$ $= \Span_{\kappa(\mm)}\{1,\,\partial_x,\,\partial_y,\,\partial_y^2,\,\partial_x^2 + \partial_x\partial_y\}$ $$I_{\mm}[\Lambda] = ((x-y)^3, (y^2+1)^2 + 4(x-y)^2, (y^2+1)(x^2-2xy-1)-4(x-y)^2)$$

## Some properties

$I$ is an arbitrary ideal
$D_{\kappa(\mm)}[I] = D_{\kappa(\mm)}[I_\mm \cap R]$

## Some properties

$I$ is an arbitrary ideal
$D_{\kappa(\mm)}[I] = D_{\kappa(\mm)}[I_\mm \cap R]$

$I$ is an ideal if and only if $D_\mm[I]$ is a local dual space
$\Lambda$ is a local dual space if and only if $I_\mm[\Lambda]$ is an ideal.

## Some properties

$I$ is an arbitrary ideal
$D_{\kappa(\mm)}[I] = D_{\kappa(\mm)}[I_\mm \cap R]$

$I$ is an ideal if and only if $D_\mm[I]$ is a local dual space
$\Lambda$ is a local dual space if and only if $I_\mm[\Lambda]$ is an ideal.

$$I \subseteq J \implies D_\mm[I] \supseteq D_\mm[J] \qquad \Lambda \subseteq \Xi \implies I_\mm[\Lambda] \supseteq I_\mm[\Xi]$$

## Some properties

$I$ is an arbitrary ideal
$D_{\kappa(\mm)}[I] = D_{\kappa(\mm)}[I_\mm \cap R]$

$I$ is an ideal if and only if $D_\mm[I]$ is a local dual space
$\Lambda$ is a local dual space if and only if $I_\mm[\Lambda]$ is an ideal.

$$I \subseteq J \implies D_\mm[I] \supseteq D_\mm[J] \qquad \Lambda \subseteq \Xi \implies I_\mm[\Lambda] \supseteq I_\mm[\Xi]$$

If $I$, $J$ ideals, $\Lambda, \Xi$ local dual spaces, then \begin{aligned} D_{\mm}[I \cap J] = D_{\mm}[I] + D_{\mm}[J] && D_{\mm}[I + J] = D_{\mm}[I] \cap D_{\mm}[J] \\ I_{\mm}[\Lambda \cap \Xi] = I_{\mm}[\Lambda] + I_{\mm}[\Xi] && I_{\mm}[\Lambda + \Xi] = I_{\mm}[\Lambda] \cap D_{\mm}[\Xi] \end{aligned}

## Some properties

$I$ is an arbitrary ideal
$D_{\kappa(\mm)}[I] = D_{\kappa(\mm)}[I_\mm \cap R]$

$I$ is an ideal if and only if $D_\mm[I]$ is a local dual space
$\Lambda$ is a local dual space if and only if $I_\mm[\Lambda]$ is an ideal.

$$I \subseteq J \implies D_\mm[I] \supseteq D_\mm[J] \qquad \Lambda \subseteq \Xi \implies I_\mm[\Lambda] \supseteq I_\mm[\Xi]$$

If $I$, $J$ ideals, $\Lambda, \Xi$ local dual spaces, then \begin{aligned} D_{\mm}[I \cap J] = D_{\mm}[I] + D_{\mm}[J] && D_{\mm}[I + J] = D_{\mm}[I] \cap D_{\mm}[J] \\ I_{\mm}[\Lambda \cap \Xi] = I_{\mm}[\Lambda] + I_{\mm}[\Xi] && I_{\mm}[\Lambda + \Xi] = I_{\mm}[\Lambda] \cap D_{\mm}[\Xi] \end{aligned}

$$D_\mm[\mm^k] = \{ D \in W_{\kappa(\mm)} \colon \deg(D) < k \}$$

## Non maximal primes

Let $\pp \subset R = \KK[x_1,\dotsc,x_n]$ be a prime.
Let $\tt = \{x_{i_1}, \dotsc, x_{i_d}\}$ denote a maximal set of algebraically independent variables in $R/\pp$,
let $\yy = \{x_{j_1}, \dotsc, x_{j_c}\}$ denote the rest.
Relabel the variables so that $R = \KK[\tt, \yy]$.

Denote by $\cdot^\ttt$ the localization at the multiplicatively closed set $\KK[\tt] \setminus \{0\}$.

$$R^\ttt = \KK(\tt)[\yy]$$

$\pp^\ttt$ is a maximal ideal in $R^\ttt$

$\kappa(\pp^\ttt) = R^\ttt/\pp^\ttt$ is isomorphic to the residue field $\kappa(\pp) = (R/\pp)_\pp$

$$R^\ttt = \KK(\tt)[\yy]$$

$\pp^\ttt$ is a maximal ideal in $R^\ttt$

$\kappa(\pp^\ttt) = R^\ttt/\pp^\ttt$ is isomorphic to the residue field $\kappa(\pp) = (R/\pp)_\pp$

Study the local dual space $$D_{\pp^\ttt}[I^\ttt] \subseteq W_{\kappa(\pp^\ttt)}$$

Warning: if $D \in W_{\kappa(\pp^\ttt)}$, then $$D = \sum_\alpha c_\alpha \partial_\yy^\alpha$$ where $c_\alpha \in \kappa(\pp)$.
There are no $\partial_\tt$ variables!

### Theorem

$\pp \in R$ prime, $\tt$ a maximal set of independent variables over $\pp$.
There are bijections
• $\pp$-primary ideals $I \subset R$ $\leftrightarrow$ finite dimensional local dual spaces $\Lambda \subseteq W_{\kappa(\pp^\ttt)}$
• $\pp$-closed ideals $I \subset R$ $\leftrightarrow$ local dual spaces $\Lambda \subseteq W_{\kappa(\pp^\ttt)}$,

given by \begin{aligned} I &\mapsto D_{\pp^\ttt}[I^\ttt] \\ \Lambda &\mapsto I_{\pp^\ttt}[\Lambda] \cap R \end{aligned}

## Example

$R = \QQ[t,x,y]$
$I = (x^2, y-tx)$ is a $\pp = (x,y)$-primary ideal.

The set $\{t\}$ is a maximal set of independent variables.
$\kappa(\pp) = \QQ(t)$

$$D_{\pp^\ttt}[I^\ttt] = \Span_{\kappa(\pp)} \left\{ 1,~\partial_y,~\partial_y^2 - \frac{2}{t}\partial_x,~\partial_y^3+\frac{6}{t}\partial_x\partial_y \right\}$$

$= \Span_{\kappa(\pp)} \left\{ 1,~\partial_y,~t\partial_y^2 - 2\partial_x,~t\partial_y^3+6\partial_x\partial_y \right\}$

### Definition

Let $I$ be $\pp$-primary, $\tt$ a maximal independent set.
A set of Noetherian operators is a finite set $\DD \subseteq W_{\kappa(\pp^\ttt)}$ such that $$\Span_{\kappa(\pp)} \DD = D_{\pp^\ttt}[I^\ttt]$$

### Theorem

$\DD$ is a set of Noetherian operators for $I$ if and only if $$I = \{ f \in R \colon D \bullet f = 0 \text{ for all } D \in \DD \}$$

## Example

$I = (y^4, xy^3, x^3y^2)$ $= (y^2) \cap (x^3,\,y^4,\,x^3y^2)$

Noetherian operators:
$\pp_1 = (y), \DD = \{1,\partial_y\}$

$\pp_2 = (x,y)$ \begin{aligned}\DD = \{& 1, \partial_x, \partial_x^2, \\ & \partial_y, \partial_y\partial_x, \partial_y\partial_x^2, \\ & \partial_y^2, \partial_y^2\partial_x, \partial_y^2\partial_x^2, \partial_y^3 \} \end{aligned}

## Example

$I = (y^4, xy^3, x^3y^2)$ $= (y^2) \cap (x^3,\,y^4,\,x^3y^2)$

Noetherian operators:
$\pp_1 = (y), \DD = \{1,\partial_y\}$

$\pp_2 = (x,y)$ \begin{aligned}\DD = \{& 1, \partial_x, \partial_x^2, \\ & \partial_y, \partial_y\partial_x, \partial_y\partial_x^2, \\ & \partial_y^2, \partial_y^2\partial_x, \partial_y^2\partial_x^2, \partial_y^3 \} \end{aligned}

$D_{(x,y)}[I]$ $D_{(x,y)}[(y^2)]$

## Example

$I = (y^4, xy^3, x^3y^2)$ $= (y^2) \cap (x^3,\,y^4,\,x^3y^2)$

Noetherian operators:
$\pp_1 = (y), \DD = \{1,\partial_y\}$

$\pp_2 = (x,y)$ \begin{aligned}\DD = \{& \partial_y^2, \partial_y^2\partial_x, \partial_y^2\partial_x^2, \partial_y^3 \} \end{aligned}

$D_{(x,y)}[I]$ $D_{(x,y)}[(y^2)]$

# Arbitrary ideals

Let $I \subseteq R$, $\pp$ an associated prime, $\tt$ a maximal set of independent variables.

Let $J = I_\pp \cap R$ be the $\pp$-closure of $I$.

$$J = Q \cap (J \colon \pp^\infty)$$

$$D_{\pp^\ttt}[J^\ttt] = D_{\pp^\ttt}[Q^\ttt] + D_{\pp^\ttt}[(J \colon \pp^\infty)^\ttt]$$

$$D_{\pp^\ttt}[I^\ttt] = D_{\pp^\ttt}[Q^\ttt] + D_{\pp^\ttt}[(I \colon \pp^\infty)^\ttt]$$

$\implies$ $\frac{D_{\pp^\ttt}[I^\ttt]}{D_{\pp^\ttt}[(I \colon \pp^\infty)^\ttt]}$ is a finite dimensional $\kappa(\pp)$-vector space.

# Arbitrary ideals

Let $I \subseteq R$, $\pp$ an associated prime, $\tt$ a maximal set of independent variables.

### Definition

The excess dual space of $I$ at $\pp$ is the finite dimensional $\kappa(\pp)$-vector space $\frac{D_{\pp^\ttt}[I^\ttt]}{D_{\pp^\ttt}[(I \colon \pp^\infty)^\ttt]}$.

### Definition

Let $I \subseteq R$ be an ideal.
For each $\pp \in \Ass(I)$ let $\tt_\pp$ be a maximal set of independent variables over $\pp$.

A differential primary decomposition is a list of triples $$\{(\pp, \tt_\pp, \DD_\pp) \}_{\pp \in \Ass(I)}$$ such that the images of $\DD_\pp \in W_{\kappa(\pp^\ttp)}$ span the excess dual space.

### Definition

Let $I \subseteq R$ be an ideal.
For each $\pp \in \Ass(I)$ let $\tt_\pp$ be a maximal set of independent variables over $\pp$.

A differential primary decomposition is a list of triples $$\{(\pp, \tt_\pp, \DD_\pp) \}_{\pp \in \Ass(I)}$$ such that the images of $\DD_\pp \in W_{\kappa(\pp^\ttp)}$ span the excess dual space.

### Proposition

$I = \{ f \in R \colon D \bullet f = 0 \text{ for all } D \in \DD_{\mathfrak{p}},\, \pp \in \Ass(I) \}$

### Definition

Let $I \subseteq R$ be an ideal.
For each $\pp \in \Ass(I)$ let $\tt_\pp$ be a maximal set of independent variables over $\pp$.

A differential primary decomposition is a list of triples $$\{(\pp, \tt_\pp, \DD_\pp) \}_{\pp \in \Ass(I)}$$ such that the images of $\DD_\pp \in W_{\kappa(\pp^\ttp)}$ span the excess dual space.

### Proposition

$I = \{ f \in R \colon D \bullet f = 0 \text{ for all } D \in \DD_{\mathfrak{p}},\, \pp \in \Ass(I) \}$

### Theorem

$\{(\pp, \tt_\pp, \DD_\pp) \}_{\pp \in \Ass(I)}$ is a differential primary decomposition if and only if, for each $\pp \in \Ass(I)$ $$I_\pp \cap R = \{ f \in R \colon D \bullet f = 0 \text{ for all } D \in \DD_{\mathfrak{q}},\, \pp \supseteq {\mathfrak{q}} \in \Ass(I) \}$$

## Example

Let $I = (x(y-z), x^2z, x^3)$: a plane $x=0$ with an embedded line $x=y-z = 0$ with a further embedded point at the origin.

$$\begin{Bmatrix} ((x), & \{y,z\}, & \{1\}), \\ ((x,y-z), & \{z\}, & \{\partial_x\}), \\ ((x,y,z), & \emptyset, & \{\partial_x^2\}) \end{Bmatrix}$$


Macaulay2, version 1.19.1.1
i1 : needsPackage "NoetherianOperators";
i2 : R = QQ[x,y];
i3 : I = ideal(x*y-x*z,x^2*z,x^3);
i4 : differentialPrimaryDecomposition I

o4 = {{ideal x, {| 1 |}}, {ideal (y - z, x), {| dx |}}, {ideal (z, y, x), {| dx^2 |}}}

o4 : List


## Example

Let $I = (x(y-z), x^2z, x^3)$: a plane $x=0$ with an embedded line $x=y-z = 0$ with a further embedded point at the origin.

$$\begin{Bmatrix} ((x), & \{y,z\}, & \{1\}), \\ ((x,y-z), & \{z\}, & \{\partial_x\}), \\ ((x,y,z), & \emptyset, & \{\partial_x^2\}) \end{Bmatrix}$$


Macaulay2, version 1.19.1.1
i1 : needsPackage "NoetherianOperators";
i2 : R = QQ[x,y];
i3 : I = ideal(x*y-x*z,x^2*z,x^3);
i4 : differentialPrimaryDecomposition I

o4 = {{ideal x, {| 1 |}}, {ideal (y - z, x), {| dx |}}, {ideal (z, y, x), {| dx^2 |}}}

o4 : List


J. Chen, Y. Cid-Ruiz, M. Härkönen, R. Krone, and A. Leykin, Noetherian operators in Macaulay2, 2021. arXiv:2101.01002.
J. Chen, M. Härkönen, R. Krone, and A. Leykin, “Noetherian operators and primary decomposition,” J. Symbolic Comput., vol. 110, 2022.

# Modules

Everything translates to $R$-submodules $U \subseteq R^k$!

# Modules

$\begin{gathered} \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 g}{\partial x \partial y} = 0 \\ \frac{\partial^2 f }{\partial y \partial z } + \frac{\partial^2 g}{\partial z^2} = 0 \\ \frac{\partial^3 f }{\partial x^2 \partial z } + \frac{\partial^3 g}{\partial z \partial y \partial w} = 0 \end{gathered}$

Everything translates to $R$-submodules $U \subseteq R^k$!

If $D \in W_{\kappa(\mm)}^k$, $f \in R^k$, then

$D \bullet f = \sum_{i=1}^k D_i \bullet f_i$

\begin{aligned} U &\mapsto D_{\pp^\ttt}[U^\ttt] \subseteq (W_{\kappa(\pp^\ttt)})^k \\ \Lambda &\mapsto I_{\pp^\ttt}[\Lambda] \cap R^k \end{aligned}

## Example

Let $$U = \im_R \begin{bmatrix}0 & x^2 & xy^2 \\ x & 0 & -2y^2 \end{bmatrix}$$

## Example

Let $$U = \im_R \begin{bmatrix}0 & x^2 & xy^2 \\ x & 0 & -2y^2 \end{bmatrix}$$


i1 : needsPackage "NoetherianOperators";
i2 : R = QQ[x,y];
i3 : U = image matrix {{0, x^2, x*y^2}, {x, 0, -2*y^2}}

o3 = image | 0 x2 xy2  |
| x 0  -2y2 |

2
o3 : R-module, submodule of R

i4 : differentialPrimaryDecomposition U

o4 = {{ideal x, {| 1 |, | 2dx |}}, {ideal (y, x), {| dx |, |  0 |}}}
| 0 |  |  1  |                    |  0 |  | dy |


## Example

Let $$U = \im_R \begin{bmatrix}0 & x^2 & xy^2 \\ x & 0 & -2y^2 \end{bmatrix}$$


i1 : needsPackage "NoetherianOperators";
i2 : R = QQ[x,y];
i3 : U = image matrix {{0, x^2, x*y^2}, {x, 0, -2*y^2}}

o3 = image | 0 x2 xy2  |
| x 0  -2y2 |

2
o3 : R-module, submodule of R

i4 : differentialPrimaryDecomposition U

o4 = {{ideal x, {| 1 |, | 2dx |}}, {ideal (y, x), {| dx |, |  0 |}}}
| 0 |  |  1  |                    |  0 |  | dy |


$\left[\begin{smallmatrix}f \\ g \end{smallmatrix} \right] \in R^2$ belongs to $U$ if and only if

• $f$ and $2\frac{\partial f}{\partial x} + g$ vanish on the line $x = 0$
• $\frac{\partial f}{\partial x}$ and $\frac{\partial g}{\partial y}$ vanish at the point $x = y = 0$.

## Example

Let $$U = \im_R \begin{bmatrix}0 & x^2 & xy^2 \\ x & 0 & -2y^2 \end{bmatrix}$$


i1 : needsPackage "NoetherianOperators";
i2 : R = QQ[x,y];
i3 : U = image matrix {{0, x^2, x*y^2}, {x, 0, -2*y^2}}

o3 = image | 0 x2 xy2  |
| x 0  -2y2 |

2
o3 : R-module, submodule of R

i4 : differentialPrimaryDecomposition U

o4 = {{ideal x, {| 1 |, | 2dx |}}, {ideal (y, x), {| dx |, |  0 |}}}
| 0 |  |  1  |                    |  0 |  | dy |


$\left[\begin{smallmatrix}f \\ g \end{smallmatrix} \right] \in R^2$ belongs to $U$ if and only if

• $f$ and $2\frac{\partial f}{\partial x} + g$ vanish on the line $x = 0$
• $\frac{\partial f}{\partial x}$ and $\frac{\partial g}{\partial y}$ vanish at the point $x = y = 0$.

J. Chen and Y. Cid-Ruiz, “Primary decomposition of modules: A computational differential approach,” Journal of Pure and Applied Algebra, 2022.
R. Ait El Manssour, M. Härkönen, and B. Sturmfels, “Linear PDE with constant coefficients,” Glasgow Mathematical Journal, First View, 2021.

# PDE

Let $R = \CC[x_1,\dotsc,x_n] = \CC[\partial_{z_1}, \dotsc, \partial_{z_n}]$.

A $\ell \times k$ matrix with entries in $R$ describes a system of $\ell$ linear homogeneous PDE with constant coefficients

Its solution is an unknown function
$v(z_1,\dotsc,z_n) \colon \RR^n \to \CC^k$

# PDE

Let $R = \CC[x_1,\dotsc,x_n] = \CC[\partial_{z_1}, \dotsc, \partial_{z_n}]$.

A $\ell \times k$ matrix with entries in $R$ describes a system of $\ell$ linear homogeneous PDE with constant coefficients

Its solution is an unknown function
$v(z_1,\dotsc,z_n) \colon \RR^n \to \CC^k$

### Example

$\ell = 3, k = 2$

$M = \begin{bmatrix} 0 & x_1 \\ x_1^2 & 0 \\ x_1 x_2^2 & -2x_2^2 \end{bmatrix}$

$M \bullet v = 0$ means finding $v(z_1,z_2) \colon \RR^2 \to \CC^2$ such that $$\begin{gathered} \partial_{z_1} \bullet v_2 = 0 \\ \partial_{z_1}^2 \bullet v_1 = 0 \\ \partial_{z_1}\partial_{z_2}^2 \bullet v_1 - 2 \partial_{z_2}^2 \bullet v_2 = 0 \end{gathered}$$

Let $\ff$ be an $R$-module of functions, $M \in R^{\ell \times k}$.

The solution space $$\Sol_\ff(M) = \{v \in \ff^k \colon M \bullet v = 0 \}$$ is an $R$-module.

It only depends on the module $U = \im_R M^T \subseteq R^k$

## $\ff = C_c^\infty$

#### Theorem (Malgrange 1960)

The set of compactly supported smooth functions is an injective $R$-module

## $\ff = C_c^\infty$

#### Theorem (Malgrange 1960)

The set of compactly supported smooth functions is an injective $R$-module

$$R^{k'} \xrightarrow{S} R^k \xrightarrow{M} R^\ell$$

$$R^{k'} \otimes_R C_c^\infty \xrightarrow{S \otimes 1} R^k \otimes_R C_c \xrightarrow{M \otimes 1} R^\ell \otimes_R C_c^\infty$$

$$(C_c^\infty)^{k'} \xrightarrow{S} (C_c^\infty)^k \xrightarrow{M} (C_c^\infty)^\ell$$

$$\Sol_{C_c^\infty}(M) = \im_{C_c^\infty}(S) = \{ S \bullet u \colon u \in (C_c^\infty)^{k'} \}$$

## $\ff = C^\infty$ or $\DD'$

### Example

$v'''(z) - 3v''(z) + 4 = 0$

Characteristic polynomial
$x^3 - 3x^2 - 4 = 0$

Characteristic polynomial
$(x-2)^2(x+1) = 0$

Solutions
$e^{2z}, ze^{2z}$
$e^{-z}$

Local dual spaces \begin{aligned} x=2&: & \langle 1, \partial_x \rangle\\ x=-1&: & \langle 1 \rangle \end{aligned}

## $\ff = C^\infty$ or $\DD'$

### Example

$v'''(z) - 3v''(z) + 4 = 0$

Characteristic polynomial
$(x-2)^2(x+1) = 0$

Solutions
$e^{2z}, ze^{2z}$
$e^{-z}$

Local dual spaces \begin{aligned} x=2&: & \langle 1, \partial_x \rangle\\ x=-1&: & \langle 1 \rangle \end{aligned}

### Theorem

Let $U \subseteq R^k$ be an $R$-submodule, $\pp$ a prime.
The operator $D = D(x,\partial_x) \in D_{\pp^\ttt}[I^\ttt]$ if and only if $$D(x_0, z) \exp(x_0^T \cdot z) \in \Sol_\ff (U)$$ for (almost) all $x_0 \in V(\pp)$.

### Theorem (improved Ehrenpreis-Palamodov)

Let $\{(\pp, \tt_\pp, \DD_\pp)\}_{\pp \in \Ass(R^k/U)}$ be a differential primary decomposition for the $R$-submodule $U \subseteq R$. All distributional solutions $u \in \Sol_{\DD'}(U)$ are of the form $$u(z) = \sum_{\pp \in \Ass(R^k/U)} \sum_{D \in \DD_\pp} \int_{V(\pp)} D(x,z) \exp(x^T \cdot z)\,\mathrm{d}\mu_{\pp,D}(x)$$ for a suitable set of measures $\mu_{\pp,D}$.

## Example

$$M = \begin{bmatrix} 0 & x_1 \\ x_1^2 & 0 \\ x_1 x_2^2 & -2x_2^2 \end{bmatrix}$$ $$U = \im_R M^T$$ Find $v \in \Sol_{\DD'}(M)$

## Example

$$M = \begin{bmatrix} 0 & x_1 \\ x_1^2 & 0 \\ x_1 x_2^2 & -2x_2^2 \end{bmatrix}$$ $$U = \im_R M^T$$ Find $v \in \Sol_{\DD'}(M)$


i1 : needsPackage "NoetherianOperators";
i2 : R = QQ[x,y];
i3 : U = image matrix {{0, x^2, x*y^2}, {x, 0, -2*y^2}}

o3 = image | 0 x2 xy2  |
| x 0  -2y2 |

2
o3 : R-module, submodule of R

i4 : solvePDE U

o4 = {{ideal x, {| 1 |, | 2dx |}}, {ideal (y, x), {| dx |, |  0 |}}}
| 0 |  |  1  |                    |  0 |  | dy |


## Example

$$M = \begin{bmatrix} 0 & x_1 \\ x_1^2 & 0 \\ x_1 x_2^2 & -2x_2^2 \end{bmatrix}$$ $$U = \im_R M^T$$ Find $v \in \Sol_{\DD'}(M)$


i1 : needsPackage "NoetherianOperators";
i2 : R = QQ[x,y];
i3 : U = image matrix {{0, x^2, x*y^2}, {x, 0, -2*y^2}}

o3 = image | 0 x2 xy2  |
| x 0  -2y2 |

2
o3 : R-module, submodule of R

i4 : solvePDE U

o4 = {{ideal x, {| 1 |, | 2dx |}}, {ideal (y, x), {| dx |, |  0 |}}}
| 0 |  |  1  |                    |  0 |  | dy |


\begin{aligned} v(z_1,z_2) =& \int_{V(x_1)} \big(\begin{smallmatrix} 1 \\ 0 \end{smallmatrix}\big) e^{x_1z_1 + x_2z_2}\,\mathrm{d}\mu_1(x) + \int_{V(x_1)} \big(\begin{smallmatrix} 2z_1 \\ 1 \end{smallmatrix}\big) e^{x_1z_1 + x_2z_2}\,\mathrm{d}\mu_2(x) \\ &+ \int_{V(x_1,x_2)} \big(\begin{smallmatrix} z_1 \\ 0 \end{smallmatrix}\big) e^{x_1z_1 + x_2z_2}\,\mathrm{d}\mu_3(x) + \int_{V(x_1,x_2)} \big(\begin{smallmatrix} 0 \\ z_2 \end{smallmatrix}\big) e^{x_1z_1 + x_2z_2}\,\mathrm{d}\mu_4(x) \end{aligned}

\begin{aligned} v(z_1,z_2) =& \int_{V(x_1)} \big(\begin{smallmatrix} 1 \\ 0 \end{smallmatrix}\big) e^{x_1z_1 + x_2z_2}\,\mathrm{d}\mu_1(x) + \int_{V(x_1)} \big(\begin{smallmatrix} 2z_1 \\ 1 \end{smallmatrix}\big) e^{x_1z_1 + x_2z_2}\,\mathrm{d}\mu_2(x) \\ &+ \big(\begin{smallmatrix} z_1 \\ 0 \end{smallmatrix}\big)c_3 + \int_{V(x_1,x_2)} \big(\begin{smallmatrix} 0 \\ z_2 \end{smallmatrix}\big) e^{x_1z_1 + x_2z_2}\,\mathrm{d}\mu_4(x) \end{aligned}

\begin{aligned} v(z_1,z_2) =& \int_{V(x_1)} \big(\begin{smallmatrix} 1 \\ 0 \end{smallmatrix}\big) e^{x_1z_1 + x_2z_2}\,\mathrm{d}\mu_1(x) + \int_{V(x_1)} \big(\begin{smallmatrix} 2z_1 \\ 1 \end{smallmatrix}\big) e^{x_1z_1 + x_2z_2}\,\mathrm{d}\mu_2(x) \\ &+ \big(\begin{smallmatrix} z_1 \\ 0 \end{smallmatrix}\big)c_3 + \big(\begin{smallmatrix} 0 \\ z_2 \end{smallmatrix}\big)c_4 \end{aligned}

\begin{aligned} v(z_1,z_2) =& \int_{\CC} \big(\begin{smallmatrix} 1 \\ 0 \end{smallmatrix}\big) e^{0z_1+x_2z_2}\,\mathrm{d}\mu_1(x_2) + \int_{\CC} \big(\begin{smallmatrix} 2z_1 \\ 1 \end{smallmatrix}\big) e^{0z_1+x_2z_2}\,\mathrm{d}\mu_2(x_2) \\ &+ \big(\begin{smallmatrix} z_1 \\ 0 \end{smallmatrix}\big)c_3 + \big(\begin{smallmatrix} 0 \\ z_2 \end{smallmatrix}\big)c_4 \end{aligned}

\begin{aligned} v(z_1,z_2) =& \int_{\CC} \big(\begin{smallmatrix} 1 \\ 0 \end{smallmatrix}\big) e^{x_2z_2}\,\mathrm{d}\mu_1(x_2) + \int_{\CC} \big(\begin{smallmatrix} 2z_1 \\ 1 \end{smallmatrix}\big) e^{x_2z_2}\,\mathrm{d}\mu_2(x_2) \\ &+ \big(\begin{smallmatrix} z_1 \\ 0 \end{smallmatrix}\big)c_3 + \big(\begin{smallmatrix} 0 \\ z_2 \end{smallmatrix}\big)c_4 \end{aligned}

\begin{aligned} v(z_1,z_2) =& \big(\begin{smallmatrix} 1 \\ 0 \end{smallmatrix}\big) \int_{\CC} e^{x_2z_2}\,\mathrm{d}\mu_1(x_2) + \big(\begin{smallmatrix} 2z_1 \\ 1 \end{smallmatrix}\big) \int_{\CC} e^{x_2z_2}\,\mathrm{d}\mu_2(x_2) \\ &+ \big(\begin{smallmatrix} z_1 \\ 0 \end{smallmatrix}\big)c_3 + \big(\begin{smallmatrix} 0 \\ z_2 \end{smallmatrix}\big)c_4 \end{aligned}

\begin{aligned} v(z_1,z_2) =& \big(\begin{smallmatrix} 1 \\ 0 \end{smallmatrix}\big) \phi_1(z_2) + \big(\begin{smallmatrix} 2z_1 \\ 1 \end{smallmatrix}\big) \phi_2(z_2) \\ &+ \big(\begin{smallmatrix} z_1 \\ 0 \end{smallmatrix}\big)c_3 + \big(\begin{smallmatrix} 0 \\ z_2 \end{smallmatrix}\big)c_4 \end{aligned}

\begin{aligned} v(z_1,z_2) = \begin{pmatrix} \phi_1(z_2) + 2z_1 \phi_2(z_2) + c_3 z_1 \\ \phi_2(z_2) + c_4z_2 \end{pmatrix} \end{aligned} for some univariate functions $\phi_1,\phi_2 \colon \RR \to \CC$ and complex values $c_3,c_4$.

## Example

$$M = \begin{bmatrix} 0 & x_1 \\ x_1^2 & 0 \\ x_1 x_2^2 & -2x_2^2 \end{bmatrix}$$ $$U = \im_R M^T$$ Find $v \in \Sol_{\DD'}(M)$


i1 : needsPackage "NoetherianOperators";
i2 : R = QQ[x,y];
i3 : U = image matrix {{0, x^2, x*y^2}, {x, 0, -2*y^2}}

o3 = image | 0 x2 xy2  |
| x 0  -2y2 |

2
o3 : R-module, submodule of R

i4 : solvePDE U

o4 = {{ideal x, {| 1 |, | 2dx |}}, {ideal (y, x), {| dx |, |  0 |}}}
| 0 |  |  1  |                    |  0 |  | dy |


\begin{aligned} v(z_1,z_2) = \begin{pmatrix} \phi_1(z_2) + 2z_1 \phi_2(z_2) + c_3 z_1 \\ \phi_2(z_2) + c_4z_2 \end{pmatrix} \end{aligned} for some univariate functions $\phi_1,\phi_2 \colon \RR \to \CC$ and complex values $c_3,c_4$.

M. Härkönen, J. Hirsch, and B. Sturmfels, Making waves, 2021. arXiv:2111.14045.

# Constant rank operators

Let $R = \KK[x_1,\dotsc,x_n]$, where $\KK = \RR$ or $\CC$.
Let $\ff = \DD'$, the space of distributions.
Let $M \in R^{\ell \times k}$ have homogeneous rows.

We say that $M$ has $\KK$-constant rank $r$ if $\rank_\KK M(x) = r$ for all $x \in \KK^m \setminus \{0\}$.

### Theorem (controllable-uncontrollable decomposition)

Let $M \in R^{\ell \times k}$ and let $S$ be the syzygy matrix of $M$.
Then there is an operator $M_1$ such that $$\Sol_\ff(M) = \im_\ff S + \Sol_\ff(M_1)$$ where $\Sol_\ff(M_1)$ contains no compactly supported distributions.

$$\Sol_\ff(M) = \im_\ff S + \Sol_\ff(M_1) \\ v = S\bullet u + w$$

$$\Sol_\ff(M) = \im_\ff S + \Sol_\ff(M_1) \\ v = S\bullet u + w$$

### Theorem

If $M$ has $\KK$-constant rank (where $\KK$ is either $\CC$ or $\RR$), then $M_1$ is $\KK$-elliptic.

$$\Sol_\ff(M) = \im_\ff S + \Sol_\ff(M_1) \\ v = S\bullet u + w$$

### Theorem

If $M$ has $\KK$-constant rank (where $\KK$ is either $\CC$ or $\RR$), then $M_1$ is $\KK$-elliptic.

We say that $M$ is $\KK$-elliptic if $\ker_\KK M(x) = \{0\}$ for all $x \in \KK^m \setminus \{0\}$.

$$\Sol_\ff(M) = \im_\ff S + \Sol_\ff(M_1) \\ v = S\bullet u + w$$

### Theorem

If $M$ has $\KK$-constant rank (where $\KK$ is either $\CC$ or $\RR$), then $M_1$ is $\KK$-elliptic.

We say that $M$ is $\KK$-elliptic if $\ker_\KK M(x) = \{0\}$ for all $x \in \KK^m \setminus \{0\}$.

If $M$ is $\RR$-elliptic, solutions to $M \bullet w = 0$ are smooth.
If $M$ is $\CC$-elliptic, solutions to $M \bullet w = 0$ are polynomials.

$$\Sol_\ff(M) = \im_\ff S + \Sol_\ff(M_1) \\ v = S\bullet u + w$$

### Theorem

If $M$ has $\KK$-constant rank (where $\KK$ is either $\CC$ or $\RR$), then $M_1$ is $\KK$-elliptic.

We say that $M$ is $\KK$-elliptic if $\ker_\KK M(x) = \{0\}$ for all $x \in \KK^m \setminus \{0\}$.

If $M$ is $\RR$-elliptic, solutions to $M \bullet w = 0$ are smooth.
If $M$ is $\CC$-elliptic, solutions to $M \bullet w = 0$ are polynomials.

M. Härkönen, B. Raiță, and L. Nicklasson, Syzygies, constant rank, and beyond, 2021. arXiv:2112.12663.

# Conclusion

### Algebra

• Ideals and modules: generators vs differential primary decomposition
• Numerical methods

### Analysis

• Analytic intuition for solving algebra problems
• Examples and counterexamples in analysis

### Algebra $\cap$ Analysis

• Interesting algebraic objects as PDE
• Interesting PDE as algebraic objects

# Conclusion

Homework: Solve the PDE

$\begin{gathered} \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 g}{\partial x \partial y} = 0 \\ \frac{\partial^2 f }{\partial y \partial z } + \frac{\partial^2 g}{\partial z^2} = 0 \\ \frac{\partial^3 f }{\partial x^2 \partial z } + \frac{\partial^3 g}{\partial z \partial y \partial w} = 0 \end{gathered}$

Hint:

needsPackage "NoetherianOperators";
viewHelp(solvePDE)

### Algebra

• Ideals and modules: generators vs differential primary decomposition
• Numerical methods

### Analysis

• Analytic intuition for solving algebra problems
• Examples and counterexamples in analysis

### Algebra $\cap$ Analysis

• Interesting algebraic objects as PDE
• Interesting PDE as algebraic objects

BREAK

# Waves

## Motivation

Let $$M = \begin{bmatrix} x_1 & x_2 & x_3 \\ x_2 & x_1 & x_4 \\ x_3 & x_4 & x_1 \end{bmatrix}$$


i9 : R = QQ[x_1,x_2,x_3,x_4];

i10 : M = matrix{{x_1, x_2, x_3},{x_2, x_1, x_4},{x_3,x_4,x_1}};

3       3
o10 : Matrix R  <--- R

i11 : U = image transpose M;


# Waves

## Motivation

Let $$M = \begin{bmatrix} x_1 & x_2 & x_3 \\ x_2 & x_1 & x_4 \\ x_3 & x_4 & x_1 \end{bmatrix}$$


i9 : R = QQ[x_1,x_2,x_3,x_4];

i10 : M = matrix{{x_1, x_2, x_3},{x_2, x_1, x_4},{x_3,x_4,x_1}};

3       3
o10 : Matrix R  <--- R

i11 : U = image transpose M;

i17 : solvePDE U

3      2      2                2
o17 = {{ideal(x  - x x  - x x  + 2x x x  - x x ), {| -x_1^2x_2x_3+x_1x_2^2x_4+x_1x_3^2x_4-x_2x_3x_4^2 |}}}
1    1 2    1 3     2 3 4    1 4    |  x_1x_2^2x_3-x_1^2x_2x_4-x_2x_3^2x_4+x_1x_3x_4^2 |
|  x_1x_2x_3^2-x_1^2x_3x_4-x_2^2x_3x_4+x_1x_2x_4^2 |

o17 : List


# Waves

## Motivation

Let $$M = \begin{bmatrix} x_1 & x_2 & x_3 \\ x_2 & x_1 & x_4 \\ x_3 & x_4 & x_1 \end{bmatrix}$$

$$\phi(z_1,z_2,z_3,z_3) = \int_V \begin{bmatrix} -x_1^2x_2x_3+x_1x_2^2x_4+x_1x_3^2x_4-x_2x_3x_4^2 \\ x_1x_2^2x_3-x_1^2x_2x_4-x_2x_3^2x_4+x_1x_3x_4^2 \\ x_1x_2x_3^2-x_1^2x_3x_4-x_2^2x_3x_4+x_1x_2x_4^2 \end{bmatrix} e^{x_1z_1 + \dotsb + x_4z_4} \,\mathrm{d}\mu(x)$$

$$V = V(x_1^3 - x_1x_2^2 - x_1x_3^2 - x_1x_4^2 + 2x_2x_3x_4)$$

# Waves

## Motivation

Let $$M = \begin{bmatrix} x_1 & x_2 & x_3 \\ x_2 & x_1 & x_4 \\ x_3 & x_4 & x_1 \end{bmatrix}$$

$$\phi(z_1,z_2,z_3,z_3) = \int_V \begin{bmatrix} -x_1^2x_2x_3+x_1x_2^2x_4+x_1x_3^2x_4-x_2x_3x_4^2 \\ x_1x_2^2x_3-x_1^2x_2x_4-x_2x_3^2x_4+x_1x_3x_4^2 \\ x_1x_2x_3^2-x_1^2x_3x_4-x_2^2x_3x_4+x_1x_2x_4^2 \end{bmatrix} e^{x_1z_1 + \dotsb + x_4z_4} \,\mathrm{d}\mu(x)$$

$$V = V(x_1^3 - x_1x_2^2 - x_1x_3^2 - x_1x_4^2 + 2x_2x_3x_4)$$

$$\pi = \Span_\RR\left\{ \begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \\ 1 \end{bmatrix}\right\} \subset V$$

# Waves

## Motivation

Let $$M = \begin{bmatrix} x_1 & x_2 & x_3 \\ x_2 & x_1 & x_4 \\ x_3 & x_4 & x_1 \end{bmatrix}$$

$$\phi(z_1,z_2,z_3,z_3) = \int_\pi \begin{bmatrix} -x_1^2x_2x_3+x_1x_2^2x_4+x_1x_3^2x_4-x_2x_3x_4^2 \\ x_1x_2^2x_3-x_1^2x_2x_4-x_2x_3^2x_4+x_1x_3x_4^2 \\ x_1x_2x_3^2-x_1^2x_3x_4-x_2^2x_3x_4+x_1x_2x_4^2 \end{bmatrix} e^{x_1z_1 + \dotsb + x_4z_4} \,\mathrm{d}\mu(x)$$

$$\pi = \Span_\RR\left\{ \begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \\ 1 \end{bmatrix}\right\} \subset V$$

# Waves

## Motivation

Let $$M = \begin{bmatrix} x_1 & x_2 & x_3 \\ x_2 & x_1 & x_4 \\ x_3 & x_4 & x_1 \end{bmatrix}$$

$$\phi(z_1,z_2,z_3,z_3) = \int_{\CC^2} \begin{bmatrix} -x_1^2x_2x_3+x_1x_2^2x_4+x_1x_3^2x_4-x_2x_3x_4^2 \\ x_1x_2^2x_3-x_1^2x_2x_4-x_2x_3^2x_4+x_1x_3x_4^2 \\ x_1x_2x_3^2-x_1^2x_3x_4-x_2^2x_3x_4+x_1x_2x_4^2 \end{bmatrix} e^{x_1z_1 + \dotsb + x_4z_4} \,\mathrm{d}\mu(x)$$

$$\pi = \Span_\RR\left\{ \begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \\ 1 \end{bmatrix}\right\} \subset V$$

# Example

$$\mm \xrightarrow{\pp} R^k$$