# solvePDE(Module) -- solve linear systems of PDE with constant coefficients

## Synopsis

• Function: solvePDE
• Usage:
solvePDE U
• Inputs:
• U, , a submodule of a free module, or a matrix, or an ideal
• Outputs:

## Description

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.