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Even if the Fourier symbols of two constant rank differential operators havethe same nullspace for each non-trivial phase space variable, the nullspaces ofthose differential operators might differ by an infinite dimensional space.Under the natural condition of constant rank over $\mathbb{C}$, we establishthat the equality of nullspaces on the Fourier symbol level already implies theequality of the nullspaces of the differential operators in $\mathscr{D}'$modulo polynomials of a fixed degree. In particular, this condition allows tospeak of natural annihilators within the framework of complexes of differentialoperators. As an application, we establish a Poincar\'{e}-type lemma fordifferential operators of constant complex rank in two dimensions.

Natural annihilators and operators of constant rank over $\mathbb{C}$