The Jacobian Conjecture, a Reduction of the Degree to the Quadratic Case

TL;DR

This paper presents a new reduction technique for the Jacobian Conjecture, showing that the problem can be simplified to the quadratic case through variable elimination and additional parameters, supported by algebraic and quantum field theory proofs.

Contribution

It introduces a novel reduction method for the Jacobian Conjecture to the quadratic case, involving partial variable elimination and a new parameter, with proofs in algebraic and quantum field theoretical frameworks.

Findings

Reduction of the Jacobian Conjecture to quadratic case.

Introduction of a parameter n' for partial variable elimination.

Algebraic and quantum field theory proofs of the reduction.

Abstract

The Jacobian Conjecture states that any locally invertible polynomial system in C^n is globally invertible with polynomial inverse. C. W. Bass et al. (1982) proved a reduction theorem stating that the conjecture is true for any degree of the polynomial system if it is true in degree three. This degree reduction is obtained with the price of increasing the dimension n. We prove here a theorem concerning partial elimination of variables, which implies a reduction of the generic case to the quadratic one. The price to pay is the introduction of a supplementary parameter 0<n'< n, parameter which represents the dimension of a linear subspace where some particular conditions on the system must hold. We first give a purely algebraic proof of this reduction result and we then expose a distinct proof, in a Quantum Field Theoretical formulation, using the intermediate field method.