Legendre Transform, Hessian Conjecture and Tree Formula

Let $\phi$ be a polynomial over $K$ (a field of characteristic 0) such that the Hessian of $\phi$ is a nonzero constant. Let $\bar\phi$ be the formal Legendre Transform of $\phi$. Then $\bar\phi$ is well-defined as a formal power series over $K$. The Hessian Conjecture introduced here claims that $\bar\phi$ is actually a polynomial. This conjecture is shown to be true when $K=\bb{R}$ and the Hessian matrix of $\phi$ is either positive or negative definite somewhere. It is also shown to be equivalent to the famous Jacobian Conjecture. Finally, a tree formula for $\bar\phi$ is derived; as a consequence, the tree inversion formula of Gurja and Abyankar is obtained.