Polynomial solutions of the complex homogeneous Monge-Ampère equation (Q1884402)

It is proved that if \(P\) is a convex homogeneous polynomial on \({\mathbb C}^n\) such that \(u=\log P\) is plurisubharmonic and satisfies the complex homogeneous Monge-Ampère equation \((dd^cu)^n=0\), then \(P\) is a homogeneous polynomial of bi-degree \((k,k)\). This partially answers the question of D. Burns if a positive homogeneous polynomial on \({\mathbb C}^n\) should have bi-degree \((k,k)\). The essence of Burns's problem is to understand the singularities of Monge-Ampère foliations and the author's approach is to construct a foliation by entire curves exploiting Lempert's results on Kobayashi geodesics in a strictly convex ball.