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The strong real Jacobian conjecture was that a real polynomial map with a nowhere vanishing Jacobian determinant has a smooth global inverse. That is equivalent to asking whether such a map is topologically a proper map, in which case it is a covering map of a simply connected manifold, hence invertible. Sergey Pinchuk constructed two variable ...
If it is true, the Jacobian conjecture would be a variant of the inverse function theorem for polynomials. It states that if a vector-valued polynomial function has a Jacobian determinant that is an invertible polynomial (that is a nonzero constant), then it has an inverse that is also a polynomial function. It is unknown whether this is true ...
In other words, if the Jacobian determinant is not zero at a point, then the function is locally invertible near this point. The (unproved) Jacobian conjecture is related to global invertibility in the case of a polynomial function, that is a function defined by n polynomials in n variables. It asserts that, if the Jacobian determinant is a non ...
In algebra the Dixmier conjecture, asked by Jacques Dixmier in 1968, [1] is the conjecture that any endomorphism of a Weyl algebra is an automorphism. Tsuchimoto in 2005, [2] and independently Belov-Kanel and Kontsevich in 2007, [3] showed that the Dixmier conjecture is stably equivalent to the Jacobian conjecture.
The strong real Jacobian conjecture was that a real polynomial map with a nowhere vanishing Jacobian determinant has a smooth global inverse. That is equivalent to asking whether such a map is topologically a proper map, in which case it is a covering map of a simply connected manifold, hence invertible.
In mathematics, the Abel–Jacobi map is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety.In Riemannian geometry, it is a more general construction mapping a manifold to its Jacobi torus.
Jacobian conjecture. Keller asked this as a question in 1939, and in the next few years there were several published incomplete proofs, including 3 by B. Segre, but Vitushkin found gaps in many of them. The Jacobian conjecture is (as of 2016) an open problem, and more incomplete proofs are regularly announced.
Ott-Heinrich Keller. Eduard Ott-Heinrich Keller (22 June 1906 in Frankfurt – 5 December 1990 in Halle) was a German mathematician who worked in the fields of geometry, topology and algebraic geometry.