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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.
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.
The absolute value of the Jacobian determinant at p gives us the factor by which the function f expands or shrinks volumes near p; this is why it occurs in the general substitution rule. The Jacobian determinant is used when making a change of variables when evaluating a multiple integral of a function over a region within its domain. To ...
The same terminology applies. A regular solution is a solution at which the Jacobian is full rank (). A singular solution is a solution at which the Jacobian is less than full rank. A regular solution lies on a k-dimensional surface, which can be parameterized by a point in the tangent space (the null space of the Jacobian).
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 ...
Jacobian conjecture: if a polynomial mapping over a characteristic-0 field has a constant nonzero Jacobian determinant, then it has a regular (i.e. with polynomial components) inverse function. Manin conjecture on the distribution of rational points of bounded height in certain subsets of Fano varieties
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.
Smale's problems is a list of eighteen unsolved problems in mathematics proposed by Steve Smale in 1998 [1] and republished in 1999. [2] Smale composed this list in reply to a request from Vladimir Arnold, then vice-president of the International Mathematical Union, who asked several mathematicians to propose a list of problems for the 21st century.