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An Optimization Approach to Jacobian Conjecture · arXivDesk

2002.10249Feb 19, 2020A research article about Jacobian Conjecture in Math

An Optimization Approach to Jacobian Conjecture

Jiang Liu

General Mathematics

Abstract

Let $n\geq 2$ and $\mathbb K$ be a number field of characteristic $0$ . Jacobian Conjecture asserts for a polynomial map $\mathcal P$ from $\mathbb K ^n$

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to itself, if the determinant of its Jacobian matrix is a nonzero constant in

\mathbb K

then the inverse

\mathcal P^{-1}

exists and is also a polynomial map. This conjecture was firstly proposed by Keller in 1939 for

\mathbb K ^n=\mathbb C^2

and put in Smale's 1998 list of Mathematical Problems for the Next Century. This study is going to present a proof for the conjecture. Our proof is based on Drużkowski Map and Hadamard's Diffeomorphism Theorem, and additionally uses some optimization idea.

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