Polynomial solution of the complex homogeneous Monge-Ampère equation

ABSTRACT: In the paper it is shown that a convex positive homogeneous polynomial on C^n such that u = log P is plurisubharmonic and satisfying (dd^c u)^n on C^n-{0} is necessarily a homogeneous polynomial of bidegree (k, k). This answer positively to a question of D. Burns for convex polynomials. The proof is based on a prolongation argument for foliations in complex geodesics through a given point of convex domains.