A homogeneous system of $l$ linear partial differential equations for a function $\psi \colon \RR^n \to \CC^k$ is encoded by a $(k \times l)$ matrix with entries in a polynomial ring. For example, if $n=4, k=2, l=3$, the PDE system $$ \frac{\partial^2 \psi_1}{\partial z_1 \partial z_3} + \frac{\partial^2 \psi_2}{\partial z_1^2} = \frac{\partial^2 \psi_1}{\partial z_1 \partial z_2} + \frac{\partial^2 \psi_2}{\partial z_2^2} = \frac{\partial^3 \psi_1}{\partial z_1^2 \partial z_2} + \frac{\partial^3 \psi_2}{\partial z_1^2 \partial z_4} =0 $$ is encoded by the matrix $$ M = \begin{bmatrix} \partial_{1} \partial_{3} & \partial_{1} \partial_{2} & \partial_{1}^2 \partial_{2}\\ \partial_{1}^2 & \partial_{2}^2 & \partial_{1}^2 \partial_{4} \end{bmatrix}, $$ or more precisely, by the submodule $U$ of $(K[\partial_1,\partial_2,\partial_3,\partial_4])^2$ generated by the columns of $M$.
By the Ehrenpreis-Palamodov fundamental principle, solutions to such PDE are of the form $$ \phi(\mathbf z) = \sum_{i = 1}^s \sum_{j=1}^{m_i} \int_{V_i} B_{i,j}(\mathbf{x}, \mathbf{z}) e^{\mathbf{x}^t \mathbf{z}} \mu_{i,j}(\mathbb{x}) $$ where $\mu_{i,j}$ are complex valued measures.
The function solvePDE computes the algebraic varieties $V_i$ and Noetherian multipliers $B_{i,j}(\mathbf x, \mathbf z)$. The input is either a matrix $M$ or a module $U$, where the $\partial_i$ is replaced by $x_i$. The output is a list of $s$ pairs. For the $i$th pair, the first entry is the prime ideal of $V_i$. The second entry is the list $B_{i,1}, \dotsc, B_{i,m_j}$ of vectors of polynomials in $2n$ variables, where the symbol $\mathbf{z}$ is replaced by the symbol $\mathbf{\mathtt{d}x}$.
i1 : R = QQ[x_1..x_4] o1 = R o1 : PolynomialRing |
i2 : M = matrix{{x_1*x_3, x_1*x_2, x_1^2*x_2}, {x_1^2, x_2^2, x_1^2*x_4}}
o2 = | x_1x_3 x_1x_2 x_1^2x_2 |
| x_1^2 x_2^2 x_1^2x_4 |
2 3
o2 : Matrix R <--- R
|
i3 : U = image M
o3 = image | x_1x_3 x_1x_2 x_1^2x_2 |
| x_1^2 x_2^2 x_1^2x_4 |
2
o3 : R-module, submodule of R
|
i4 : sols = solvePDE M
o4 = {{ideal x , {| 1 |}}, {ideal (x , x ), {| -x_1 |}}, {ideal (x , x ), {|
1 | 0 | 4 2 | x_3 | 3 2 |
------------------------------------------------------------------------
1 |}}, {ideal (x , x ), {| x_2dx_1 |}}, {ideal (x , x ), {| 0 |, | 0
0 | 3 1 | -1 | 2 1 | 1 | | dx_1
------------------------------------------------------------------------
2 2
|, | 0 |, | 0 |}}, {ideal (x - x x , x x - x x , x - x x ),
| | dx_2 | | dx_1dx_2 | 2 1 4 1 2 3 4 1 2 3
------------------------------------------------------------------------
{| -x_4 |}}}
| x_2 |
o4 : List
|
i5 : netList sols
+-----------------------------------------+-----------------------------------------+
o5 = |ideal x |{| 1 |} |
| 1 | | 0 | |
+-----------------------------------------+-----------------------------------------+
|ideal (x , x ) |{| -x_1 |} |
| 4 2 | | x_3 | |
+-----------------------------------------+-----------------------------------------+
|ideal (x , x ) |{| 1 |} |
| 3 2 | | 0 | |
+-----------------------------------------+-----------------------------------------+
|ideal (x , x ) |{| x_2dx_1 |} |
| 3 1 | | -1 | |
+-----------------------------------------+-----------------------------------------+
|ideal (x , x ) |{| 0 |, | 0 |, | 0 |, | 0 |}|
| 2 1 | | 1 | | dx_1 | | dx_2 | | dx_1dx_2 | |
+-----------------------------------------+-----------------------------------------+
| 2 2 | |
|ideal (x - x x , x x - x x , x - x x )|{| -x_4 |} |
| 2 1 4 1 2 3 4 1 2 3 | | x_2 | |
+-----------------------------------------+-----------------------------------------+
|
This output reveals that the general solution to the example system above consists of nine summands, one of which is $$ \phi(\mathbb z) = \int_{V(x_3,x_1)} \begin{bmatrix} z_1 x_2 \\ -1 \end{bmatrix} e^{x_1 z_1 + x_2 z_2 + x_3 z_3 + x_4 z_4} \, d\mu(x_1,x_2,x_3,x_4) $$
The total number of Noetherian multipliers is equal to the arithmetic multiplicity of the module $U$.
i6 : amult U == sum(sols / last / (l -> #l)) o6 = true |
Note that the output of solvePDE can be interpreted as a differential primary decomposition.