Verification of the Jacobian Conjecture for $d$-linear maps in two variables

TL;DR

This paper verifies the Jacobian Conjecture for all d-linear maps in two variables, showing most inverse coefficients satisfy the Jacobian condition and deriving related algebraic expressions.

Contribution

It provides a proof of the Jacobian Conjecture for d-linear maps in two variables and generalizes a Cayley-Hamilton theorem proof to obtain key algebraic expressions.

Findings

Almost all inverse coefficients lie in the Jacobian ideal

Derived explicit expressions for elements in the ideal

Generalized a bijective Cayley-Hamilton proof

Abstract

For any integer $d \geq 1$, we verify the Jacobian Conjecture for a $d$-linear map in two variables. We prove that almost all the coefficients of the formal inverse are in the ideal specified by the Jacobian condition. We find expressions for certain elements in terms of the generators of this ideal. To obtain these expressions, we generalize a bijective proof of the Cayley-Hamilton theorem in two ways.