Natural annihilators and operators of constant rank over C

Even if the Fourier symbols of two constant rank differential operators have the same nullspace for each non-trivial phase space variable, the nullspaces of those differential operators might differ by an infinite-dimensional space. Under the natural condition of constant rank over C, we establish that the equality of nullspaces on the Fourier symbol level already implies the equality of the nullspaces of the differential operators in D′ modulo polynomials of a fixed degree. In particular, this condition allows one to speak of natural annihilators within the framework of complexes of differential operators.