In mathematics, the Jacobian conjecture is a conjecture concerning polynomials in several variables that states that if a polynomial function from an n {\displaystyle n} -dimensional space to itself has a Jacobian determinant that is a non-zero constant, then the function has a polynomial inverse. The case n = 2 {\displaystyle n=2} (two variables), also called the plane Jacobian conjecture or planar Jacobian conjecture, is the only case that remains unresolved as of 2026. The case n = 1 {\displaystyle n=1} is trivially true, since the derivative of a polynomial is a nonzero constant only if the degree of the polynomial is 1 , {\displaystyle 1,} and linear polynomial functions are invertible. On July 19, 2026, Levent Alpöge presented an explicit counterexample in three variables which he credited to Claude Fable 5, which disproves the conjecture for n > 2 {\displaystyle n>2} . The correctness of the counterexample is easy to verify with any computer algebra system. It has not been revealed, however, how it was found. Nevertheless, it led some mathematicians to elaborate on the mathematical reasons and the implications of the existence of the counterexample.
History The Jacobian conjecture was originally formulated without being named in two dimensions by Ludwig Kraus in 1884, who gave a flawed proof in the same paper. Later, the modern version of the Jacobian conjecture in n {\displaystyle n} dimensions was formulated by Ott-Heinrich Keller in 1939 for the case of polynomials with integer coefficients. Arno van den Essen has claimed that Keller only talked about the two-dimensional case; however, Keller in fact did talk about the general n {\displaystyle n} -dimensional case. For nearly a century, Keller was considered to be the first person to formulate the conjecture, until a 2025 search of the zbMATH database revealed that the two-dimensional case had already been stated by Kraus. The conjecture was unnamed in either of Kraus's or Keller's original papers. It is unclear who coined the name "Jacobian conjecture", but this was because it involves the Jacobian determinant, itself named after the German mathematician Carl Gustav Jacob Jacobi. The earliest known published use of the name occurs in Masayoshi Miyanishi's 1973 paper, where it refers to the n {\displaystyle n} -dimensional conjecture. Tzuong-Tsieng Moh later recalled that, after Oscar Zariski pointed out at a seminar at Purdue University in the late 1960s that the assertion remained unproved, "we decided to call it the Jacobian Conjecture". Alexander Borisov later attributed the coinage specifically to Shreeram Abhyankar, whose 1977 lecture notes treated the two-dimensional problem and presented results that he had obtained in 1970–71. The conjecture, also called Jacobian problem, was subsequently widely publicized by Abhyankar as an example of a difficult question in algebraic geometry that can be understood using little beyond a knowledge of calculus. The Jacobian conjecture is number 16 in Stephen Smale's 1998 list of Mathematical Problems for the Next Century. According to Alexander Borisov, the conjecture in two dimensions has been especially well studied in the literature on the conjecture. Historically, some mathematicians, such as Shreeram Abhyankar, Tzuong-Tsieng Moh, and Yitang Zhang, have used the term Jacobian conjecture or Jacobian problem to refer to only the conjecture in two dimensions. There have been a large number of false proofs of the conjecture in two dimensions, some of them published. Arno van den Essen in 1997, Tzuong-Tsieng Moh in 1998, and Edward Formanek in 2011 all hypothesized that the conjecture could be true in two dimensions and false in the general case. After the counterexample in three dimensions was discovered in 2026 only the conjecture in two dimensions remains open.
Formulation of the conjecture Let n > 1 {\displaystyle n>1} be a fixed integer and consider polynomials f 1 , … , f n {\displaystyle f_{1},\dots ,f_{n}} in variables X 1 , … , X n {\displaystyle X_{1},\dots ,X_{n}} with coefficients in a field K {\displaystyle \mathbb {K} } . Then we define a vector-valued function F : K n → K n {\displaystyle F:\mathbb {K} ^{n}\to \mathbb {K} ^{n}} by setting:
F ( X 1 , … , X n ) = ( f 1 ( X 1 , … , X n ) , … , f n ( X 1 , … , X n ) ) {\displaystyle F(X_{1},\dots ,X_{n})=(f_{1}(X_{1},\dots ,X_{n}),\dots ,f_{n}(X_{1},\dots ,X_{n}))}
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