Legendre Transform, Hessian Conjecture and Tree Formula

Guowu Meng

Let $\varphi$ be a polynomial over $K$ (a field of characteristic 0) such that the Hessian of $\varphi$ is a nonzero constant. Let $\bar{\varphi }$ be the formal Legendre Transform of $\varphi$. Then $\bar{\varphi }$…