"The Jacobian Conjecture is a criterion that relates the invertibility of polynomial mappings of C^n to a condition on their Jacobian matrix… and it remains unproven for dimensions greater than one." "The case n = 2 has been documented by a great many authors, but remains unproven." "Despite this, no proof for mappings of degree 3 has been found, so the conjecture remains unproven."

**Jacobian conjecture**: Let k be an algebraically closed field of characteristic zero, n ≥ 2 and φ: k^n → k^n a (regular) endomorphism of k^n with constant Jacobian (the determinant of the Jacobian matrix, which is in this polynomial case algebraically defined). Then φ is a regular automorphism, i.e. has a polynomially defined inverse. The conjecture has been stated by Keller in 1939 and is still open. There were many failed attempts to prove the Jacobian conjecture, especially for n = 2; there are also some reductions to special cases. The Jacobian conjecture is also equivalent to the Dixmier conjecture: every endomorphism of the r-th Weyl algebra A_{r,k} over k is an automorphism for all r.

In mathematics, the Jacobian conjecture is a famous unsolved problem concerning polynomials in several variables. As of 2018, it has not been proven, even for the two-variable case. Van den Essen provides evidence that the conjecture may be false for large numbers of variables. The Jacobian conjecture is notorious for the large number of published and unpublished proofs that turned out to contain subtle errors.

"The Jacobian conjecture in the plane, first stated by Keller (1939)… has been an open problem since Keller (1939)." "The Jacobian conjecture is one of Smale's problems. There have been at least five published incorrect proofs and many incorrect attempts over the years." "In November 2004, Hochster (2004) sent an email announcing a new proof by Carolyn Dean. However, this proof unfortunately contained an error as well."

This book deals with the development on polynomial rings in the last two decades since the publication of “Polynomial Automorphisms” by Arno van den Essen. It provides an updated overview of results related to polynomial automorphisms and the Jacobian conjecture, including reductions of the conjecture and partial results, but does not claim a complete proof of the Jacobian conjecture. The conjecture is still presented as open, with emphasis on its difficulty and the many partial approaches.

The Jacobian Conjecture is a long-standing open problem in affine algebraic geometry. It asserts the invertibility of a polynomial map F : Cn → Cn when the Jacobian determinant det JF is a nonzero constant. In this paper we improve the algebraic methods of Abhyankar describing the shape of the support of possible counterexamples. Our results show that any counterexample must satisfy very restrictive conditions on its support.

The Jacobian conjecture raised by Keller in 1939, claiming that polynomial maps F : k^n → k^n, k being a field of characteristic zero, whose Jacobian determinant is a nonzero constant, are bijective with polynomial inverse, is still open. In this paper we present a counterexample to a folk real Jacobian conjecture, showing a non-injective polynomial map from R^2 to R^2 with Jacobian determinant bounded away from 0. This does not contradict Keller’s conjecture, which is formulated over fields of characteristic zero and requires a constant Jacobian determinant.

"The Jacobian conjecture can be reduced to the consideration of polynomial maps F:C^n→C^n of the special form F=X−H… In that case, it asserts that if |J(F)|, the Jacobian determinant of F, is identically 1, then F is bijective, with a polynomial inverse." "In all these five cases, the stated conjectures are equivalent to the Jacobian conjecture if they are true for a fixed d≥3 and all values of n≥2." This paper treats the conjecture as unsolved and focuses on equivalent formulations and reductions rather than a proof.

Most of us know the Jacobian conjecture. We know that the sentence "For all n, J(3,n)" implies the sentence "For all d,n, J(d,n)". In other words, the Jacobian conjecture has been reduced to degree 3. We also know that, for any fixed n, J(3,n) is provably true or provably false. This boils down to the completeness of the theory of algebraically closed fields of characteristic zero. There's no way to rule out a priori that the Jacobian conjecture is undecidable (in your favorite axiomatic system).

In this work we used tools of dynamical systems and algebraic geometry to give a new characterization of the Jacobian conjecture for polynomial local homeomorphisms in R^2. Our characterization is equivalent to the Jacobian conjecture in the real plane but it does not resolve the conjecture; it reformulates it in terms of dynamical properties of the associated vector fields. The Jacobian conjecture in the real plane therefore remains an open problem, even in dimension two, and we provide a new perspective rather than a proof.

"The Jacobian Conjecture on polynomial self-maps of C^n (or more general field of characteristic 0) is open even in the case n=2." "Some special cases are known such as the case of quadratic polynomials." This discussion explicitly treats the conjecture as an open problem and analyzes its logical/arithmetic form, not as a solved theorem.

We do know that proving the JC is hard and the history of its "proofs" is littered with failed attempts by strong professional mathematicians. In fact, we do not know if the JC is hard or not; for all what we know, some smart undergraduate can simply write a formula of a degree 3 polynomial vector-function in several complex variables that will be a counter-example to this conjecture. However, we do know that proving the JC is hard and the history of its "proofs" is littered with failed efforts (two of them, Segre and Gröbner, are quite famous).

"The Jacobian Conjecture is one of the most well-known open problems in algebraic geometry." "It now seems that a proof has been found by Carolyn Dean of the University of Michigan, for the case of polynomials in two complex variables…" "Update: Someone wrote in with a comment to another post pointing out that Dean has found a hole in her proof." This blog post records both the claim of a proof and the subsequent acknowledgement of a gap, leaving the conjecture still open.

"When it comes to the progress on JC, several reductions have been made…" "Currently, the research has developed in a somewhat different direction, as a result of some new conjectures by W. Zhao that are either equivalent to the Jacobian conjecture, or imply it." This answer describes known reductions and related conjectures, and treats the Jacobian conjecture explicitly as an unsolved problem with partial progress rather than a proven statement.

Based on the reduction of degree in polynomial mappings and some known results in algebraic geometry, by introducing the Brouwer degree, a tool from differential topology, algebraic topology and algebraic geometry, we completely prove the Generalized Jacobian conjecture in the field of real numbers, which implies the Generalized complex Jacobian conjecture. Also, for the strong real Jacobian conjecture, we present a newly sufficient and necessary condition. [Comments: 10 pages, globally injective on origin, homotopical invariance.]

"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." This historical study confirms that Kraus's attempted proof was flawed and that the conjecture remains a conjecture.

The Jacobian conjecture, a problem from algebraic geometry which is concerned with the invertibility of polynomial mappings in variables over a field of characteristic zero, remains open. In this paper we study random planar trees and derive combinatorial structures related to polynomial mappings, obtaining new insights into the problem but not a proof of the conjecture. Our results show that certain combinatorial models can capture aspects of the Jacobian condition, suggesting further directions but leaving the conjecture unresolved.

"Real Jacobian Conjecture: if the Jacobian of the map of R^n→R^n vanishes at nowhere, then the map is globally invertible." This document discusses real variants and related statements, presenting them as conjectures or problems rather than settled theorems, which indicates ongoing research activity around Jacobian-type global invertibility questions.

This study presents a non-injective polynomial map from R^2 to R^2 with Jacobian determinant bounded away from 0. This provides a counterexample to a folk real Jacobian conjecture which claimed that such maps must be injective. The example does not have constant Jacobian determinant, so it does not contradict the classical Jacobian conjecture of Keller, but it shows that certain strengthened real versions are false.

The Reddit discussion focuses on a recent 12‑page paper claiming to prove the Jacobian conjecture, criticizing it as "mostly just recounting the basic theory of C^1 functions" and pointing out an "obvious flaw" in the alleged proof. It also notes another claimed proof by Yucai Su on arXiv which has been "withdrawn or updated over 40 times" and states that this history "does not serve to inspire confidence" in these proofs. This shows ongoing but so far unsuccessful attempts to prove the conjecture.

The Jacobian conjecture has been stated by Keller in 1939 and is still open. There were many failed attempts to prove the Jacobian conjecture. For example, it is known that the Jacobian conjecture holds iff it holds for φ a polynomial map of degree 3. There is a recent proof of related Kontsevich’s statement on automorphisms of Weyl algebra, but this concerns an equivalent formulation and does not resolve the original conjecture.