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On the origin of the Jacobian conjecture

Lázaro O. Rodríguez Díaz

TL;DR

The paper traces the origin of the plane Jacobian conjecture to Kraus's 1884 work, showing that his statement matches Keller's later formulation for $J(p,q)=1$. It reconstructs Kraus's approach based on the resultant and algebraic-function theory, linking modern results (e.g., Kaliman, Formanek, Weierstrass) to justify several steps under irreducible fibers. The key contribution is a precise reconstruction of Kraus's argument and a clear pinpointing of the flaw at infinity: the step invoking a nonzero derivative at infinity, $w'_{\alpha}(0)\neq 0$, is not valid because the relevant derivative is not defined there. This analysis clarifies that ramification at infinity remains the principal obstacle to algebro-geometric proofs of the Jacobian conjecture, connecting Kraus's ideas to contemporary methodologies.

Abstract

The Jacobian conjecture is thought to have been proposed by O. H. Keller in 1939. However, we have found that the statement of the conjecture is precisely the main result of a paper published by L. Kraus in 1884. Although the final step of Kraus's proof is flawed, the ideas he introduced anticipated approaches to the problem that would only emerge more than a century later. Interestingly, the root of Kraus's error remains the principal obstacle to algebro-geometric approaches: controlling the ramification at infinity.

On the origin of the Jacobian conjecture

TL;DR

The paper traces the origin of the plane Jacobian conjecture to Kraus's 1884 work, showing that his statement matches Keller's later formulation for . It reconstructs Kraus's approach based on the resultant and algebraic-function theory, linking modern results (e.g., Kaliman, Formanek, Weierstrass) to justify several steps under irreducible fibers. The key contribution is a precise reconstruction of Kraus's argument and a clear pinpointing of the flaw at infinity: the step invoking a nonzero derivative at infinity, , is not valid because the relevant derivative is not defined there. This analysis clarifies that ramification at infinity remains the principal obstacle to algebro-geometric proofs of the Jacobian conjecture, connecting Kraus's ideas to contemporary methodologies.

Abstract

The Jacobian conjecture is thought to have been proposed by O. H. Keller in 1939. However, we have found that the statement of the conjecture is precisely the main result of a paper published by L. Kraus in 1884. Although the final step of Kraus's proof is flawed, the ideas he introduced anticipated approaches to the problem that would only emerge more than a century later. Interestingly, the root of Kraus's error remains the principal obstacle to algebro-geometric approaches: controlling the ramification at infinity.
Paper Structure (1 section, 3 theorems, 10 equations)

This paper contains 1 section, 3 theorems, 10 equations.

Table of Contents

  1. Kraus's proof

Key Result

Theorem 1

Let $(f_1(x,y),f_2(x,y))$ be a polynomial map of $\mathbb{C}^{2}$. There is an equivalence between: Furthermore, if $(f_1,f_2)$ is invertible, then $(g_1,g_2)$ is its inverse.

Theorems & Definitions (4)

  • Theorem 1: Adjamagbo_Essen90
  • Remark 2
  • Theorem 3: Abhyankar71, McKay_Wang86
  • Theorem 4: Weierstrass_1902, Hejmej18, Garcia_Barroso-Gwozdziewic22