Comptes Rendus
Histoire des sciences et des idées - Histoire des mathématiques
On the origin of the Jacobian conjecture
[Sur l’origine de la conjecture jacobienne]
Lázaro Orlando Rodríguez Díaz1 orcid
1Instituto de Matemática, Universidade Federal do Rio de Janeiro, RJ, Brazil
Comptes Rendus. Mathématique, Volume 364 (2026), pp. 363-370

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 pense que la conjecture jacobienne a été formulée par O. H. Keller en 1939. Nous avons toutefois découvert que l’énoncé de cette conjecture correspond exactement au résultat principal d’un article publié par L. Kraus en 1884. Bien que la dernière étape de la démonstration de Kraus soit erronée, les idées qu’il a introduites anticipaient des approches du problème qui n’allaient apparaître que plus d’un siècle plus tard. Il est intéressant de noter que la source de l’erreur de Kraus reste le principal obstacle aux approches algébro-géométriques : le contrôle de la ramification à l’infini.

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DOI : 10.5802/crmath.831
Classification : 14R15, 13P15, 12E05, 14H05, 01A55
Keywords: Jacobian conjecture, Jacobian determinant, branch point, resultant, Ludwig Kraus, irreducibility
Mots-clés : Conjecture jacobienne, déterminant jacobien, point de ramification, résultant, Ludwig Kraus, irréductibilité

Lázaro Orlando Rodríguez Díaz  1

1 Instituto de Matemática, Universidade Federal do Rio de Janeiro, RJ, Brazil
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Lázaro Orlando Rodríguez Díaz. On the origin of the Jacobian conjecture. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 363-370. doi: 10.5802/crmath.831
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