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A system of polynomial equations related to the Jacobian Conjecture

Jorge A. Guccione, Juan José Guccione, Christian Valqui

Abstract

We prove that the Jacobian conjecture is false if and only if there exists a solution to a certain system of polynomial equations. We analyse the solution set of this system. In particular we prove that it is zero dimensional.

A system of polynomial equations related to the Jacobian Conjecture

Abstract

We prove that the Jacobian conjecture is false if and only if there exists a solution to a certain system of polynomial equations. We analyse the solution set of this system. In particular we prove that it is zero dimensional.

Paper Structure

This paper contains 6 sections, 20 theorems, 221 equations.

Table of Contents

  1. The Jacobian Conjecture as a system of equations
  2. Properties of solutions of $\bm{S(n,m,(\lambda_i),F_{1-n})}$
  3. The homogeneous system $\bm{S(n,m,F_{1-n})}$
  4. Presentations of the solutions of $\bm{S(n,m,Y^{m+n-1})}$
  5. A modified system and an example
  6. Acknowledgment

Key Result

Proposition 1.1

The Jacobian conjecture is false if and only if there exists a Jacobian pair $(P,Q)$, such that neither $\deg(P)$ divides $\deg(Q)$ nor $\deg(Q)$ divides $\deg(P)$.

Theorems & Definitions (50)

  • Proposition 1.1: vdE*Theorem 10.2.23
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4: vdE*page 247
  • Proposition 1.5
  • proof
  • Lemma 1.6
  • proof
  • Proposition 1.7
  • Remark 1.8
  • ...and 40 more