An Optimization Approach to Jacobian Conjecture

TL;DR

This paper claims to prove the Jacobian Conjecture for polynomial maps over number fields using Drużkowski maps, Hadamard's theorem, and optimization techniques, addressing a long-standing open problem in mathematics.

Contribution

The paper presents a proof of the Jacobian Conjecture leveraging Drużkowski maps, Hadamard's Diffeomorphism Theorem, and optimization methods, which is a novel approach to this longstanding problem.

Findings

Proof of the Jacobian Conjecture for polynomial maps over number fields.

Application of Drużkowski maps and Hadamard's Theorem in the proof.

Use of optimization techniques to establish invertibility conditions.

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.