Legendre transform, Hessian conjecture and tree formula

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