An Optimization Approach to Jacobian Conjecture

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 $\mathbb K ^n$ 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 Dru{\.{z}}kowski Map and Hadamard's Diffeomorphism Theorem, and additionally uses some optimization idea.