Polynomial Structures in Generalized Geometry

On the generalized tangent bundle of a smooth manifold, we study skew-symmetric endomorphism satisfying an arbitrary polynomial equation with real constant coefficients. We study the compatibility of these structures with the de Rham operator and the Courant-Dorfman bracket. In particular we isolate several conditions that when restricted to the motivating example of generalized almost complex structure are equivalent to the notion of integrability.