Reference request: Systems of linear PDES with constant coefficients I am looking for a reference for the following statement:
Assume that $P_1, \dots, P_k \in \mathbb R[x_1, \dots, x_m]$ and consider a system of PDEs
\begin{align}
P_i(\partial / \partial x_1, \dots, \partial / \partial x_m)u = f_i, \quad i = 1, \dots, k
\end{align}
where $u(x_1, \dots, x_m)$ is a scalar function to be found.
Then, if this system has a solution, we should obviously have
$$
\sum_{i=1}^k Q_i(\partial / \partial x_1, \dots, \partial / \partial x_m)f_i = 0
$$
for each $k$-tuple $(Q_1, \dots, Q_k)$ such that 
$$
\sum_{i=1}^k Q_iP_i = 0.
$$
The statement is that this condition is also sufficient for local solvability of our system of PDEs. More precisely,
1) if $f_1, \dots, f_k$ are $\mathrm C^\infty$ functions in a neighborhood of $x \in \mathbb R^m$, then there exists a $\mathrm C^\infty$ solution in a (possibly smaller) neighborhood of $x$;
2) if $f_1, \dots, f_k$ are analytic, then the solution is also analytic.
I suspect that this should be Ehrenpreis or Malgrange, but I was not able to find the precise statement.
Another question is whether there is a geometric way to understand this result. For instance, if $P_1, \dots, P_k$ are of degree $1$, then the statement can be deduced from the Frobenius theorem.
 A: I think what you are looking for is something like Theorem 9 in the following reference:
Ehrenpreis, Leon, A fundamental principle for systems of linear differential equations with constant coefficients, and some of its applications. (1961) Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960) pp. 161–174 Jerusalem Academic Press, Jerusalem; Pergamon, Oxford.
A: Another reference for results on solvability of linear constant coefficient PDEs is

Tarkhanov, N. N. Complexes of Differential Operators, vol. 340 of Mathematics and Its Applications (Kluwer, Dordrecht, 1995)
  http://dx.doi.org/10.1007/978-94-011-0327-5

In particular, see Secs. 1.2.2 and 1.2.4. The latter comes with extensive references on this subject. The works of Ehrenpreis, Malgrange and Palamodov get a particular mention, with final credit given to Malgrange.
A: Theorem 7.6.13 in Hörmander's book "An Introduction to Complex Analysis in Several Variables" (North-Holland, 1990, 3rd edition) contains the result you are looking for. Sections 7.6 and 7.7 of this book give a full prove of the Ehrenpreis Fundamental Principle of which solvability is a part. See Hörmander's notes at the end of chapter 7 for historical information.
