Linear PDE with constant coefficients

\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
Seminar on Nonlinear Algebra, June 11 2021

A series of motivating questions

Study the ideal $$ I = (z^2, (x-y)^2-2xz, zy(x-y)) $$

Find the points $(x,y,z)$ satisfying \begin{align*} z^2 = (x-y)^2-2xz = zy(x-y) = 0 \end{align*}

Answer: the line $z = 0$, $x = y$.

Compute a primary decomposition of $$ I = (z^2, (x-y)^2-2xz, zy(x-y)) $$

Answer: $$\small I = (z^2, (x-y)z, (x-y)^2-2yz) \\ \cap \\ (y,z^2, x^2-2xz)$$

Compute a primary decomposition of $$ I = (z^2, (x-y)^2-2xz, zy(x-y)) $$

Answer: $$\small I = (z^2, (x-y)z, (x-y)^2-2yz) \\ \cap \\ (y,z^2, x^2-2xz)$$

Triple line $z=0, x=y$,
embedded point of length 1 at the origin.

Describe $$ I = (z^2, (x-y)^2-2xz, zy(x-y)) $$ using differential equations.

Answer: the polynomial $p \in I$ iff

$p$ vanishes on the line $z = x-y = 0$
$\partial_x p$ vanishes on the line $z = x-y = 0$
$y\partial_x^2 p + \partial_z p$ vanishes on the line $z = x-y = 0$
$\partial_x^2 p + \partial_x \partial_z p$ vanishes at the origin

Solve the PDE $$ \partial_z^2 f = 0\\ ((\partial_x - \partial_y)^2 - 2\partial_x \partial_z) f = 0 \\ \partial_z \partial_y (\partial_x - \partial_y) f = 0 $$

Answer: $$ f(x,y,z) = \alpha(x+y) + \\ x\beta(x+y) + \\x^2 \gamma'(x+y) + z \gamma(x+y) +\\ c\cdot(x^2+xz) $$ where $\alpha,\beta,\gamma$ are univariate functions, $c$ a constant

Setup

$R = \mathbb{C}[\partial_{z_1},\dotsc,\partial_{z_n}]$

Elements of $R$ operate on functions $f \in \mathcal{D'}(\mathbb{R}^n)$ $f(z_1,\dotsc, z_n) \colon \mathbb{R}^n \to \mathbb{C}$

E.g. $(2\partial_1^2 - 3) \bullet f = 2\frac{\partial^2 f}{\partial z_1^2} - 3 f$.

Setup

$R = \mathbb{C}[\partial_{z_1},\dotsc,\partial_{z_n}]$

Elements of $R^k$ operate on functions $f \in $ $\mathcal{D'}(\mathbb{R}^n)^k$ $f(z_1,\dotsc, z_n) \colon \mathbb{R}^n \to$ $\mathbb{C}^k$.

\begin{align*} R^k \times \mathcal{D}'^k &\to \mathcal{D}' \\ p \bullet f &\mapsto \sum_{i=1}^k p_i \bullet f_i \end{align*}

E.g. $$ \begin{bmatrix} \partial_1\partial_2 \\ \partial_3 - 1 \end{bmatrix} \bullet \begin{bmatrix} f_1 \\ f_2 \end{bmatrix} = \partial_1 \partial_2 \bullet f_1 + (\partial_3 - 1) \bullet f_2 $$

Linear partial differential operators with constant coefficients are encoded as vectors in $R^k = \mathbb{C}[\partial_1, \dotsc, \partial_n]^k$.

A set of $l$ partial differential operators is encoded by a $k \times l$ matrix $M$ with entries in $R$.

A function $f \in \mathcal{D}'^k$ satisfies the PDE $M$ if $M_i \bullet f = 0$ for all $l$ columns $M_i$ of $M$.

Example

$$ M = \begin{bmatrix} \partial_x^2 & \partial_x\partial_y & \partial_y^2 + \partial_x\\ \partial_x+1 & \partial_y^2 & -\partial_x + 1 \end{bmatrix} $$ corresponds to the PDE $$ \frac{\partial^2 f_1}{\partial x^2} + \frac{\partial f_2}{\partial x} + f_2 = 0 \\ \frac{\partial^2 f_1}{\partial x \partial y} + \frac{\partial^2 f_2}{\partial y^2} = 0\\ \frac{\partial^2 f_1}{\partial y^2} + \frac{\partial f_1}{\partial x} - \frac{\partial f_2}{\partial x}+f_2 = 0 $$

One solution (among infinitely many) is $$ \begin{bmatrix} f_1(x,y) \\ f_2(x,y) \end{bmatrix} = \begin{bmatrix} -\cos x - \sin x \\ - \sin x \end{bmatrix} $$

Proposition

The function $f \in \mathcal{D}'^k$ satisfies the PDE $M$ if and only if $m \bullet f = 0$ for all $m \in U$, where $U$ is the submodule of $R^k$ generated by the columns of $M$.

PDE $\iff$ Matrix $\iff$ Submodule of $R^k$.

If $U \subseteq R^k$, define the $R$-module of solutions $$\mathrm{Sol}(U) := \{f \in \mathcal{D}' \mid m \bullet f =0 \, \forall m \in U \}$$

Simple algebra

Varieties

Suppose $k=1$, submodules $I \subseteq R^1$ are ideals.

Proposition

The function $e^{\xi_1 z_1 + \dotsb + \xi_n z_n} \in \mathrm{Sol}(I)$ if and only if $(\xi_1,\dotsc,\xi_n) \in V(I)$.

E.g. if $I = (\partial_3^2, (\partial_1-\partial_2)^2-2\partial_1\partial_3, \partial_3\partial_2(\partial_1-\partial_2))$, then the "pure" exponential solutions are parametrized by $$ t \mapsto e^{tz_1 + tz_2 + 0z_3} $$

A similar statement is true for modules ($k>1$). The analogue of $V(I)$ is the characteristic variety $$V_U = \bigcup_{P \in \mathrm{Ass}(R^k/U)} V(P) = V(k\text{-minors of M}) = V(\mathrm{Ann}_R(R^k/U))$$

Syzygies

Suppose $U$ is generated by the columns of matrix $M$. Let $S$ be the matrix whose image is the module of relations between rows of $M$

$$ R^l \xleftarrow{M^T} R^k \xleftarrow{S} R^{k_1}$$

Then $S^T \bullet \mathcal{D}'^{k_1} \subset \mathrm{Sol}(U)$, and

$S^T \bullet \mathcal{D}'^{k_1} = \mathrm{Sol}(U)$ if $R^k/U$ is torsion free.

Example

Let $M = \mathrm{curl} = \begin{bmatrix} -\partial_y & 0 & -\partial_z \\ \partial_x & -\partial_z & 0 \\ 0 & \partial_y & \partial_x \end{bmatrix}$

$$ R^3 \xleftarrow{\begin{bmatrix} -\partial_y & \partial_x & 0 \\ 0 & -\partial_z & \partial_y \\ -\partial_z & 0 & \partial_x \end{bmatrix}} R^3 \xleftarrow{\begin{bmatrix} \partial_x \\ \partial_y \\ \partial_z \end{bmatrix}} R^1 $$

The operator $\begin{bmatrix} \partial_x & \partial_y & \partial_z \end{bmatrix}$ is the gradient operator.

Furthermore, $R^3/\mathrm{Im}_R(M)$ is torsion-free.

Curl-free functions are precisely gradients

$\mathrm{curl} f = 0 \implies f = \nabla g$ for some function $g$.

solvePDE

Given $U \subseteq R^k$, how do we get all of $\mathrm{Sol}(U)$?

Ehrenpreis-Palamodov fundamental principle

There exist varieties $\{V_i\}$ and polynomial vectors $\{B_{i,j}(\mathbf{x},\mathbf{z})\}$ such that every solution $f \in \mathrm{Sol}(U)$ is written as an integral

$$ f(\mathbb{z}) = \sum_{i=1}^L \sum_{j=1}^{m_j} \int_{V_i} B_{i,j}(\mathbb{x},\mathbb{z}) e^{\mathbb{x} \cdot \mathbb{z}} \, d\mu_{i,j}(\mathbb{x}) $$

for some measures $\{\mu_{i,j}\}$.

$$ f(\mathbb{z}) = \sum_{i=1}^L \sum_{j=1}^{m_j} \int_{V_i} B_{i,j}(\mathbb{x},\mathbb{z}) e^{\mathbb{x} \cdot \mathbb{z}} \, d\mu_{i,j}(\mathbb{x}) $$

find the varieties $V_i$ and polynomials $B_{i,j}(\mathbf{x},\mathbf{z})$

Ehrenpreis-Palamodov:
These are the varieties corresponding to associated primes of $R^k/U$


Macaulay2, version 1.18.0.1

i1 : R = QQ[x,y,z];

i2 : M = matrix {{x^2,x*y},{x+1,y^2}}

o2 = | x2  xy |
     | x+1 y2 |

             2       2
o2 : Matrix R  <--- R

i3 : associatedPrimes coker M

o3 = {ideal y, ideal x, ideal(x*y - x - 1)}

o3 : List

find the varieties $V_i$ and polynomials $B_{i,j}(\mathbf{x},\mathbf{z})$

We call the $B_{ij}(\mathbb{x},\mathbb{z})$ Noetherian multipliers

Dual problem: Differential primary decomposition

Denote $R = \mathbb{C}[\partial_{z_1}, \dotsc, \partial_{z_n}] := \mathbb{C}[x_1,\dotsc,x_n]$. Find the differential operators $B_{i,j}(\mathbf{x}, \partial_\mathbf{x})$ such that $$ f \in U \iff B_{i,j} \bullet f \in P_i \text { for all } P_i \in \mathrm{Ass}(R^k/U)$$

We call the $B_{ij}(\mathbb{x},\partial_x)$ Noetherian operators

Cid-Ruiz, Y., & Sturmfels, B. (2021). Primary Decomposition with Differential Operators. arXiv:2101.03643.
Chen, J., & Cid-Ruiz, Y. (2021). Primary decomposition of modules: a computational differential approach. arXiv:2104.03385.

Describe $$ I = (z^2, (x-y)^2-2xz, zy(x-y)) $$ using differential equations.

Answer: the polynomial $p \in I$ iff

$p$ vanishes on the line $z = x-y = 0$
$\partial_x p$ vanishes on the line $z = x-y = 0$
$y\partial_x^2 p + \partial_z p$ vanishes on the line $z = x-y = 0$
$\partial_x^2 p + \partial_x \partial_z p$ vanishes at the origin

solvePDE

Distributed with Macaulay2 from version 1.18 in the package NoetherianOperators

Study the PDE $$ M = \begin{bmatrix} \partial_1^2 - 4\partial_1\partial_2-\partial_2 & \partial_1^2 \partial_2 - \partial_2^2 & \partial_1^3 - \partial_1 \partial_2 \\ 2\partial_1^2-2\partial_2 & 0 & 0 \end{bmatrix} $$

i.e. find $f_1(z_1,z_2)$, $f_2(z_1,z_2)$ such that \begin{align*} (\partial_1^2 - 4\partial_1\partial_2-\partial_2)f_1 + (2\partial_1^2-2\partial_2)f_2 &= 0\\ (\partial_1^2 \partial_2 - \partial_2^2)f_1 &= 0\\ (\partial_1^3 - \partial_1 \partial_2)f_1 &= 0 \end{align*}

Study the module $U \subseteq \mathbb{C}[x_1,x_2]$ generated by the columns of $$ M = \begin{bmatrix} x_1^2 - 4x_1x_2-x_2 & x_1^2 x_2 - x_2^2 & x_1^3 - x_1 x_2 \\ 2x_1^2-2x_2 & 0 & 0 \end{bmatrix} $$


Macaulay2, version 1.18.0.1

i1 : needsPackage "NoetherianOperators";

i2 : R = QQ[x_1,x_2];

i3 : U = image matrix {{x_1^2-4*x_1*x_2-x_2, x_1^2*x_2-x_2^2, x_1^3-x_1*x_2}, {2*x_1^2-2*x_2, 0, 0}}

o3 = image | x_1^2-4x_1x_2-x_2 x_1^2x_2-x_2^2 x_1^3-x_1x_2 |
           | 2x_1^2-2x_2       0              0            |

                             2
o3 : R-module, submodule of R

i4 : differentialPrimaryDecomposition U

              2
o4 = {{ideal(x  - x ), {| 0 |, |    1    |}}, {ideal (x , x ), {| -2dx_2 |}}}
              1    2    | 1 |  | x_2dx_1 |             2   1    |  dx_2  |

o4 : List

Associated primes $P_1 = (x_1^2 - x_2)$, $P_2 = (x_1,x_2)$.

Noetherian operators $B_{1,1} = (0, 1)^T$, $B_{1,2} = (1, x_2\partial_{x_1})^T$, $B_{2,1} = (-2\partial_{x_2}, \partial_{x_2})^T$.

$(p_1,p_2) \in U$ if and only if

$p_2 \in P_1$
$p_1 + x_2 \partial_{x_1} p_2 \in P_1$
$-2\partial_{x_2} p_1 + \partial_{x_2} p_2 \in P_2$

$U = U_1 \cap U_2$ is a primary decomposition, where $$ U_1 = \{p \in R^2 \mid B_{1,1} \bullet p \in P_1, B_{1,2} \bullet p \in P_1\}$$ $$ U_2 = \{p \in R^2 \mid B_{2,1} \bullet p \in P_2\}$$


Macaulay2, version 1.18.0.1

i1 : needsPackage "NoetherianOperators";

i2 : R = QQ[x_1,x_2];

i3 : U = image matrix {{x_1^2-4*x_1*x_2-x_2, x_1^2*x_2-x_2^2, x_1^3-x_1*x_2}, {2*x_1^2-2*x_2, 0, 0}}

o3 = image | x_1^2-4x_1x_2-x_2 x_1^2x_2-x_2^2 x_1^3-x_1x_2 |
           | 2x_1^2-2x_2       0              0            |

                             2
o3 : R-module, submodule of R

i4 : solvePDE U

              2
o4 = {{ideal(x  - x ), {| 0 |, |    1    |}}, {ideal (x , x ), {| -2dx_2 |}}}
              1    2    | 1 |  | x_2dx_1 |             2   1    |  dx_2  |

o4 : List

i5 : netList oo

     +--------------+--------------------+
     |       2      |                    |
o5 = |ideal(x  - x )|{| 0 |, |    1    |}|
     |       1    2 | | 1 |  | x_2dx_1 | |
     +--------------+--------------------+
     |ideal (x , x )|{| -2dx_2 |}        |
     |        2   1 | |  dx_2  |         |
     +--------------+--------------------+

Solving PDEs:
the $x_i$ variables correspond to $\partial_{z_i}$,
the $z_i$ variables correspond to $\partial_{z_i}$.

Output is the same as differentialPrimaryDecomposition!

Varieties: $V_1 = V(x_1^2 - x_2)$, $V_2 = V(x_1,x_2)$.

Noetherian multipliers $B_{1,1} = (0, 1)^T$, $B_{1,2} = (1, x_1z_1)^T$, $B_{2,1} = (-2z_2, z_2)^T$.

Solutions: \begin{align*} \begin{bmatrix} f_1(z_1,z_2) \\ f_2(z_1,z_2) \end{bmatrix} =& \int_{V_1} \begin{bmatrix} 0 \\ 1 \end{bmatrix} e^{x_1z_1 + x_2z_2} \, d\mu_{1,1}(x_1,x_2) \\ &+\int_{V_1} \begin{bmatrix} 1 \\ x_1z_1 \end{bmatrix} e^{x_1z_1 + x_2z_2} \, d\mu_{1,2}(x_1,x_2)\\ &+\int_{V_2} \begin{bmatrix} -2z_2 \\ z_2 \end{bmatrix} e^{x_1z_1 + x_2z_2} \, d\mu_{2,1}(x_1,x_2) \end{align*}

Solutions: \begin{align*} \begin{bmatrix} f_1(z_1,z_2) \\ f_2(z_1,z_2) \end{bmatrix} =& \int_{V_1} \begin{bmatrix} 0 \\ 1 \end{bmatrix} e^{x_1z_1 + x_2z_2} \, d\mu_{1,1}(x_1,x_2) \\ &+\int_{V_1} \begin{bmatrix} 1 \\ x_1z_1 \end{bmatrix} e^{x_1z_1 + x_2z_2} \, d\mu_{1,2}(x_1,x_2)\\ &+c\begin{bmatrix} -2z_2 \\ z_2 \end{bmatrix} \end{align*}

Solutions: \begin{align*} \begin{bmatrix} f_1(z_1,z_2) \\ f_2(z_1,z_2) \end{bmatrix} =& \begin{bmatrix} 0 \\ 1 \end{bmatrix} \int_{V_1} e^{x_1z_1 + x_2z_2} \, d\mu_{1,1}(x_1,x_2) \\ &+\begin{bmatrix} 1 \\ \partial_{z_1}z_1 \end{bmatrix} \int_{V_1} e^{x_1z_1 + x_2z_2} \, d\mu_{1,2}(x_1,x_2)\\ &+c\begin{bmatrix} -2z_2 \\ z_2 \end{bmatrix} \end{align*}

Making waves

A wave is a function that is constant along affine spaces

If $f \colon \mathbb{R}^\ell \to \mathbb{C}^k$ is any $n-\ell$-variate function, then $f(\xi_1 \cdot x, \dotsc, \xi_{n-\ell} \cdot x)$ is a wave for all $\xi_i \in \mathbb{R}^k$, constant on $\ell$-dimensional hyperplanes.

Suppose $M \in \mathbb{C}[\partial_1,\dotsc,\partial_n]^{k\times l}$ has entries that are homogeneous of the same degree.

The solutions are $k$-vectors

Suppose we want to find solutions of the form $$ vf(\xi_1\cdot x, \dotsc, \xi_{n-\ell} \cdot x), $$ where $v \in \mathbb{P}^{k-1}$, $\xi_i \in \mathbb{P}^{n-1}$, and $f \colon \mathbb{R}^{n-\ell} \to \mathbb{C}$

We need $$ v \in \mathcal{N}^\ell := \bigcup_{\pi \in \mathrm{Gr}(n-\ell, n) }\bigcap_{\xi \in \pi \setminus \{0\}} \mathrm{ker} M^T(\xi) $$

$ \mathcal{N}^0 \subseteq \dotsb \subseteq \mathcal{N}^{n-1}$

Let $M = \begin{bmatrix} \partial_1 & \partial_2 & \partial_3 \\ \partial_2 & \partial_1 & \partial_4 \\ \partial_3 & \partial_4 & \partial_1 \end{bmatrix}$


netList solvePDE M

+----------------------------------------+------------------------------------------------------+
|       3      2      2                2 |                                                      |
|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 | |
+----------------------------------------+------------------------------------------------------+

$$V_1 = V(\det M(\mathbf{x}))$$

$$B_{1,1}(\mathbf{x},\mathbf{z}) = \begin{bmatrix} -(x_1x_3 - x_2x_4)(x_1x_2-x_3x_4) \\ (x_2x_3-x_1x_4)(x_1x_2-x_3x_4)\\ (x_2x_3-x_1x_4)(x_1x_3-x_2x_4) \end{bmatrix} $$

Let $M = \begin{bmatrix} \partial_1 & \partial_2 & \partial_3 \\ \partial_2 & \partial_1 & \partial_4 \\ \partial_3 & \partial_4 & \partial_1 \end{bmatrix}$

Suppose $$v \in \mathcal{N}^2 := \bigcup_{\pi \in \mathrm{Gr}(2, 4) }\bigcap_{\xi \in \pi \setminus \{0\}} \mathrm{ker} M^T(\xi)$$

$0 = M^T(\xi)v $

$0 = M^T(\xi)v = \begin{bmatrix}v_1 & v_2 & v_3 & 0 \\ v_2 & v_1 & 0 & v_3 \\ v_3 & 0 & v_1 & v_2\end{bmatrix} \xi$

$0 = M^T(\xi)v = \begin{bmatrix}v_1 & v_2 & v_3 & 0 \\ v_2 & v_1 & 0 & v_3 \\ v_3 & 0 & v_1 & v_2\end{bmatrix} \xi$
$\implies \mathcal{N}^2 $is the variety corresponding to the $3 \times 3$ minors of the matrix above

$\mathcal{N}^2$ consists of six projective points, each of which is the normal of the lines going between the vertices of the elliptope

e.g. $(1,0,-1)^T \in \mathcal{N}^2$. The function $$ f(x_1,x_2,x_3,x_4) = \begin{cases} (1,0,-1)^T, & \text{ if } x_1 + x_3 = x_2 + x_4 = 0\\ (0,0,0)^T, & \text{otherwise} \end{cases} $$ satisfies the PDE \begin{gather*} \partial_1 f_1 + \partial_2 f_2 + \partial_3 f_3 = 0\\ \partial_2 f_1 + \partial_1 f_2 + \partial_4 f_3 = 0\\ \partial_3 f_1 + \partial_4 f_2 + \partial_1 f_3 = 0 \end{gather*}

A numerical future

We use the data "associated primes" $V(P_i)$ and "Noetherian multipliers" $B_{i,j}(\mathbb{x},\mathbb{z})$ to describe submodules of $\mathbb{C}[\mathbb{x}]^k$.

$V(P_i)$ are represented by witness sets

We use the data "associated primes" $V(P_i)$ and "Noetherian multipliers" $B_{i,j}(\mathbb{x},\mathbb{z})$ to describe submodules of $\mathbb{C}[\mathbb{x}]^k$.

$V(P_i)$ are represented by witness sets

$B_{ij}(\mathbf{x},\mathbf{z})$ could be represented by sets of witness solutions

If $\xi$ lies approximately on $V(P_i)$ and $B(\mathbf x, \mathbf z)$ is a Noetherian multiplier, we say that $$ B(\xi, \mathbf{z})e^{\xi \cdot z} $$ is a witness solution

We use the data "associated primes" $V(P_i)$ and "Noetherian multipliers" $B_{i,j}(\mathbb{x},\mathbb{z})$ to describe submodules of $\mathbb{C}[\mathbb{x}]^k$.

$V(P_i)$ are represented by witness sets

$B_{ij}(\mathbf{x},\mathbf{z})$ could be represented by sets of witness solutions

If $\xi$ lies approximately on $V(P_i)$ and $B(\mathbf x, \mathbf z)$ is a Noetherian multiplier, we say that $$ B(\xi, \mathbf{z})e^{\xi \cdot z} $$ is a witness solution

Numerical module membership test, numerical primary decomposition, numerical module operations: sums, intersections...

References

Härkönen, M., Hirsch, J., & Sturmfels, B. (in preparation). Noetherian Operators and Wave Cones

Ait El Manssour, R., Härkönen, M., & Sturmfels, B. (2021). Linear PDE with Constant Coefficients.

Chen, J., & Cid-Ruiz, Y. (2021). Primary decomposition of modules: a computational differential approach.

Chen, J., Härkönen, M., Krone, R., & Leykin, A. (2020). Noetherian operators and primary decomposition.