The Jacobian Conjecture as a problem in combinatorics

TL;DR

This paper approaches the Jacobian Conjecture through combinatorics, providing an inversion formula, relating it to the Grossman-Larson Algebra, and proving the conjecture for specific cases, thus offering new algebraic insights.

Contribution

It introduces a combinatorial framework and an inversion formula for the symmetric case, advancing the understanding of the Jacobian Conjecture and proving it for certain classes.

Findings

Proved the symmetric Jacobian Conjecture for F=X-H with H homogeneous and JH^3=0

Established a relation between the inversion formula and the Grossman-Larson Algebra

Proposed a combinatorial statement that could lead to a complete proof of the Jacobian Conjecture

Abstract

The Jacobian Conjecture has been reduced to the symmetric homogeneous case. In this paper we give an inversion formula for the symmetric case and relate it to a combinatoric structure called the Grossman-Larson Algebra. We use these tools to prove the symmetric Jacobian Conjecture for the case $F=X-H$ with $H$ homogeneous and $JH^{3}=0$. Other special results are also derived. We pose a combinatorial statement which would give a complete proof the Jacobian Conjecture.