  In this paper an equation means a homogeneous linear partial differential
equation in $n$ unknown functions of $m$ variables which has real or complex
polynomial coefficients. The solution set consists of all $n$-tuples of real or
complex analytic functions that satisfy the equation. For a given system of
equations we would like to characterize its Weyl closure, i.e. the set of all
equations that vanish on the solution set of the given system. It is well-known
that in many special cases the Weyl closure is equal to $B_m(F)N \cap A_m(F)^n$
where $F$ is either the field of real or complex numbers, $A_m(F)$
(respectively $B_m(F)$) consists of all linear partial differential operators
with coefficients in $F[x_1,\ldots,x_m]$ (respectively $F(x_1,\ldots,x_m)$) and
$N$ is the submodule of $A_m(F)^n$ generated by the given system. Our main
result is that this formula holds in general. In particular, we do not assume
that the module $A_m(F)^n/N$ has finite rank which used to be a standard
assumption. Our approach works also for the real case which was not possible
with previous methods. Moreover, our proof is constructive as it depends only
on the Riquier-Janet theory.
