Jacobian conjecture: coloring Abdesselam-Rivasseau model

TL;DR

This paper transforms the Jacobian Conjecture into a quantum field theory problem using the Abdesselam-Rivasseau model, demonstrating the series termination for the inverse map through colored edge analysis, which suggests JC's correctness.

Contribution

It introduces a novel approach by applying colored edge techniques in the AR model to prove the series termination related to JC.

Findings

Series for the inverse map terminates with edge coloring.

Termination implies the correctness of the Jacobian Conjecture.

Method offers a new perspective on JC via quantum field theory.

Abstract

We consider the Abdesselam-Rivasseau (AR) model turning the Jacobian Conjecture (JC) into a problem of the perturbative quantum field theory. Within the AR model, the JC inverse map is represented by a formal integral generating the tree's expansion for this map. By assigning colors to the edges in the vertex of the AR model and performing selective Gaussian integration, we show the termination of the tree's series for the inverse map. The latter implies the correctness of JC.