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Some classes satisfying the 2-dimensional Jacobian conjecture and a proof of the complex conjecture until degree 104

Thuy Nguyen

TL;DR

This work addresses the two-dimensional Jacobian conjecture by introducing a non-proper variable framework that yields explicit JC-satisfying classes in both real and complex settings. It shows how maps can be expressed as $F=(F_1,F_2)=H(u_0,\dots,u_n)$ with a non-proper variable set $\{u_i\}$, enabling classification via independence/dependence criteria. Using Newton polygon techniques and a synthesis of Abhyankar–Nagata–Appelgate-Onishi–Magnus–Nakai–Baba–Żoła\ndek–Moh results, the paper proves the complex JC for degrees up to $104$, improving the previous bound of $100$ (Moh) and clarifying degree-case analyses. It further reduces the degree-105 problem to four gcd-based cases, offering a more accessible path through classical casework and highlighting potential connections between the non-proper-variable approach and higher-degree conjectures.

Abstract

We construct a non-proper set of two variables polynomial maps and study the nowhere vanishing Jacobian condition of the Jacobian conjecture for this set. We obtain some classes of polynomial maps satisfying the 2-dimensional Jacobian conjecture for both real and complex cases. In addition, by Newton polygon technique, we prove that the complex conjecture is true until degree 104, improving Moh boundary (degree 100) since 1983.

Some classes satisfying the 2-dimensional Jacobian conjecture and a proof of the complex conjecture until degree 104

TL;DR

This work addresses the two-dimensional Jacobian conjecture by introducing a non-proper variable framework that yields explicit JC-satisfying classes in both real and complex settings. It shows how maps can be expressed as with a non-proper variable set , enabling classification via independence/dependence criteria. Using Newton polygon techniques and a synthesis of Abhyankar–Nagata–Appelgate-Onishi–Magnus–Nakai–Baba–Żoła\ndek–Moh results, the paper proves the complex JC for degrees up to , improving the previous bound of (Moh) and clarifying degree-case analyses. It further reduces the degree-105 problem to four gcd-based cases, offering a more accessible path through classical casework and highlighting potential connections between the non-proper-variable approach and higher-degree conjectures.

Abstract

We construct a non-proper set of two variables polynomial maps and study the nowhere vanishing Jacobian condition of the Jacobian conjecture for this set. We obtain some classes of polynomial maps satisfying the 2-dimensional Jacobian conjecture for both real and complex cases. In addition, by Newton polygon technique, we prove that the complex conjecture is true until degree 104, improving Moh boundary (degree 100) since 1983.

Paper Structure

This paper contains 9 sections, 21 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Non-proper variables sets of a polynomial map
  3. Construction and examples
  4. A set of non-proper variables
  5. The set of variables (\ref{['eq:p-r']}) is a set of non-proper variables
  6. The set of non-proper variables (\ref{['eq:p-r']}) with $r=1$
  7. The set of non-proper variables (\ref{['eq:p-r']}) with $n=1$ and $r \geq 2$
  8. The set of non-proper variables (\ref{['eq:p-r']}) with $n\geq 2$ and $r \geq 2$
  9. Two-dimensional complex Jacobian conjecture of degree until 104

Key Result

Proposition 2.6

Let $F=(F_1, F_2): \mathbb{K}^2 \to \mathbb{K}^2$ be a non-proper polynomial map and $\{u_0, u_1, \dots, u_n\}$ be a set of non-proper variables of $F$. If $\{u_0, u_1, \ldots, u_n\}$ is dependent then $F$ does not satisfy the Jacobian condition.

Theorems & Definitions (41)

  • Remark 2.1
  • Definition 2.2: non-proper variables sets
  • Remark 2.3
  • Definition 2.4: Dependent/Independent non-proper variables sets
  • Example 2.5: Chau
  • Proposition 2.6
  • proof
  • Corollary 2.7
  • Corollary 2.8
  • Example 2.9
  • ...and 31 more