Legendre Transform, Hessian Conjecture and Tree Formula

Abstract

Let $ϕ$ be a polynomial over $K$ (a field of characteristic 0) such that the Hessian of $ϕ$ is a nonzero constant. Let $\barϕ$ be the formal Legendre Transform of $ϕ$. Then $\barϕ$ is well-defined as a formal power series over $K$. The Hessian Conjecture introduced here claims that $\barϕ$ is actually a polynomial. This conjecture is shown to be true when $K=\bb{R}$ 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 $\barϕ$ is derived; as a consequence, the tree inversion formula of Gurja and Abyankar is obtained.