A proof of the Generalized Jacobian conjecture

TL;DR

This paper proves the Generalized Jacobian conjecture over real numbers using algebraic geometry and differential topology tools, and also provides a new necessary and sufficient condition for the strong real Jacobian conjecture.

Contribution

It offers a complete proof of the Generalized Jacobian conjecture over real numbers and introduces a new condition for the strong real Jacobian conjecture.

Findings

Complete proof of the Generalized Jacobian conjecture over real numbers

Implication of the conjecture for complex numbers

New necessary and sufficient condition for the strong real Jacobian conjecture

Abstract

Based on the reduction of degree in polynomial mappings and some known results in algebraic geometry, by introducing the Brouwer degree, a tool from differential topology, algebraic topology and algebraic geometry, we completely prove the Generalized Jacobian conjecture in the field of real numbers, which implies the Generalized complex Jacobian conjecture. Also, for the strong real Jacobian conjecture, we present a newly sufficient and necessary condition.