  We study the local invariants that a meromorphic k-differential on a Riemann
surface of genus greater or equal to zero can have for k greater or equal to 3.
These local invariants are the orders of zeros and poles, and the k-residues at
the poles. We show that for a given pattern of orders of zeros, there exists,
up to a few exceptions, a primitive holomorphic k-differential having zeros of
these orders. In the meromorphic case, in genus greater or equal to one, every
expected tuple appears as k-residues. On the other hand, some expected tuples
are not the k-residus of a $k$-differential in some remaining strata. This
happens in genus zero for a finite number (up to simultaneous scaling) of them.

Complex Variables;Algebraic Geometry;Geometric Topology