Legendre Transform, Hessian Conjecture and Tree Formula
| dc.creator | Meng, Guowu | |
| dc.date | 2003-08-28 | |
| dc.date | 2005-01-31 | |
| dc.date.accessioned | 2026-07-07T06:26:50Z | |
| dc.date.available | 2026-07-07T06:26:50Z | |
| dc.description | 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. | |
| dc.description | 9 pages, references are updated | |
| dc.identifier | https://arxiv.org/abs/math-ph/0308035 | |
| dc.identifier | http://arxiv.org/abs/math-ph/0308035 | |
| dc.identifier | Appl. Math. Lett. 19 (2006), 503-510 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/97235 | |
| dc.subject | Mathematical Physics | |
| dc.subject | Combinatorics | |
| dc.title | Legendre Transform, Hessian Conjecture and Tree Formula | |
| dc.type | text |