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Integral curves and Jacobian Conjecture

Jean-Yves Charbonnel

TL;DR

The paper tackles the Jacobian Conjecture by focusing on étale polynomial endomorphisms of $\mathbb{C}^n$ of the form $\Phi(x)=x+F^{(3)}(x)$ and leveraging Drużkowski's reduction. It develops a semialgebraic framework and a vector-field approach, introducing the integral curve flow $X_x(y)=\Phi'(y)^{-1}(x)$ and showing that maximal integral curves yield a polynomial inverse to $\Phi$, proving $\Phi$ is a polynomial automorphism when étale. Through an analysis of algebraic curves $\Gamma_x$ and their étale coverings, it establishes finiteness and rigidity properties that force the global invertibility of $\Phi$, culminating in the main theorem that étale polynomial endomorphisms are automorphisms. Consequently, the Jacobian Conjecture holds in this setting, with implications extending to automorphism results for Weyl algebras over characteristic-zero fields.

Abstract

In this note, we are interested in the Jacobian Conjecture. Following the results of L.M.~Dru$\dot{\rm z}$kowski, we consider some vector fields depending on a certain étale polynomial map. From results of semialgebraic geometry with the consideration of some integral curves of these vector fields, we deduce that an étale polynomial endomorphism is a polynomial automorphism.

Integral curves and Jacobian Conjecture

TL;DR

The paper tackles the Jacobian Conjecture by focusing on étale polynomial endomorphisms of of the form and leveraging Drużkowski's reduction. It develops a semialgebraic framework and a vector-field approach, introducing the integral curve flow and showing that maximal integral curves yield a polynomial inverse to , proving is a polynomial automorphism when étale. Through an analysis of algebraic curves and their étale coverings, it establishes finiteness and rigidity properties that force the global invertibility of , culminating in the main theorem that étale polynomial endomorphisms are automorphisms. Consequently, the Jacobian Conjecture holds in this setting, with implications extending to automorphism results for Weyl algebras over characteristic-zero fields.

Abstract

In this note, we are interested in the Jacobian Conjecture. Following the results of L.M.~Drukowski, we consider some vector fields depending on a certain étale polynomial map. From results of semialgebraic geometry with the consideration of some integral curves of these vector fields, we deduce that an étale polynomial endomorphism is a polynomial automorphism.
Paper Structure (19 sections, 32 theorems, 86 equations)

This paper contains 19 sections, 32 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Sketch of proof
  3. Notations and Definition
  4. Semialgebraic sets and maps
  5. Some results on semialgebraic sets
  6. On semialgebraic maps
  7. On semialgebraic maps
  8. Vector fields and integral curves
  9. Elementary properties
  10. On the integral curves
  11. Special integral curves
  12. On some algebraic curves
  13. On the varieties
  14. Étale covering
  15. A property of the function
  16. ...and 4 more sections

Key Result

Proposition 1.2

Let $x$ be in ${\mathbb C}^{n}$ and $\gamma _{x}$ the maximal integral curve of $X_{x}$ with initial datum $\gamma _{x}(0)=0$. Then ${\mathbb R}$ is the interval of definition of $\gamma _{x}$.

Theorems & Definitions (65)

  • Conjecture 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Definition 1.4
  • Lemma 1.5
  • proof
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 55 more