New Proofs for the Abhyankar-Gurjar Inversion Formula and the Equivalence of the Jacobian Conjecture and the Vanishing Conjecture

TL;DR

This paper presents new proofs and formulations for the Abhyankar-Gurjar inversion formula and demonstrates the equivalence of the Jacobian Conjecture with a specific case of the Vanishing Conjecture, simplifying previous approaches.

Contribution

It introduces a new formulation of the Abhyankar-Gurjar inversion formula and provides a more straightforward proof of the Jacobian Conjecture's equivalence to a Vanishing Conjecture case.

Findings

New formulation of the Abhyankar-Gurjar inversion formula

Simplified proof of the Jacobian Conjecture equivalence

Connection between the Jacobian Conjecture and Vanishing Conjecture

Abstract

We first give a new proof and also a new formulation for the Abhyankar-Gurjar inversion formula for formal maps of affine spaces. We then use the reformulated Abhyankar-Gurjar formula to give a more straightforward proof for the equivalence of the Jacobian Conjecture with a special case of the Vanishing Conjecture of (homogeneous) quadratic differential operators with constant coefficients.