On the origin of the Jacobian conjecture

TL;DR

This paper reveals that the Jacobian conjecture was actually proposed in 1884 by Kraus, whose ideas foreshadowed later approaches, but a flaw in his proof highlights ongoing challenges in controlling ramification at infinity.

Contribution

It uncovers the true origin of the Jacobian conjecture and analyzes Kraus's early ideas, emphasizing the persistent difficulty of managing ramification at infinity.

Findings

Kraus's 1884 paper contains the original statement of the Jacobian conjecture.

Kraus's proof has a flaw related to ramification at infinity.

Understanding Kraus's approach offers insights into current challenges in the conjecture.

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.