Legendre Transform, Hessian Conjecture and Tree Formula

TL;DR

This paper explores the properties of the Legendre Transform of polynomials with constant Hessian, proposing the Hessian Conjecture that such transforms are polynomials, and connects it to the Jacobian Conjecture, providing a new tree formula.

Contribution

The paper introduces the Hessian Conjecture for formal Legendre Transforms of polynomials with constant Hessian and establishes its equivalence to the Jacobian Conjecture, also deriving a novel tree formula.

Findings

Hessian Conjecture holds over real numbers with definite Hessian.

Equivalence established between Hessian Conjecture and Jacobian Conjecture.

Derived a tree formula for the formal Legendre Transform.

Abstract

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.