Polynomial cubic differentials and convex polygons in the projective plane

We construct and study a natural homeomorphism between the moduli space of polynomial cubic differentials of degree d on the complex plane and the space of projective equivalence classes of oriented convex polygons with d + 3 vertices. This map arises from the construction of a complete hyperbolic affine sphere with prescribed Pick differential, and can be seen as an analogue of the Labourie–Loftin parameterization of convex RP 2 structures on a compact surface by the bundle of holomorphic cubic differentials Teichmüller space.