Legendre Transform, Hessian Conjecture and Tree Formula

Abstract

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

9 pages, references are updated