Legendre transform, Hessian conjecture and tree formula
Legendre transform, Hessian conjecture and tree formula
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DOI：
10.1016/j.aml.2005.07.006
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发表时间：
2003-08
期刊：
影响因子：
--
通讯作者：
Guowu Meng
中科院分区：
文献类型：
--
作者：
Guowu Meng
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.
