Ludwig Kraus

Ludwig Kraus (; 9 April 1857 – 1886) was a Czech mathematician. In 1884, he published a paper giving a partial proof of the result now known as the two-variable case of the Jacobian conjecture more than half a century before Ott-Heinrich Keller proposed the conjecture in its – now disproven – general form in 1939. Kraus's priority went unrecognised until 2026. His next two papers, both titled ("On rationally invertible substitutions"), appeared in the proceedings of the Royal Bohemian Society of Sciences in Prague in 1882 and 1883. In them, he investigated pairs of integral rational functions of two variables for which the variables can be recovered from the substituted values as rational functions. published in the proceedings of the Imperial Academy of Sciences in Vienna, has as its main result the statement now called the plane (two-variable) Jacobian conjecture, namely that a polynomial map of the plane whose Jacobian determinant is a nonzero constant possesses a polynomial inverse. Kraus argued the result using the theory of the resultant and the theory of algebraic functions, but the final step of his proof is flawed. A 2026 study found that the gap, controlling the ramification at infinity, is the same obstacle that continues to block algebro-geometric approaches to the conjecture.