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In mathematics, the Jacobian conjecture is a famous unsolved problem concerning polynomials in several variables. It states that if a polynomial function from an n -dimensional space to itself has Jacobian determinant which is a non-zero constant, then the function has a polynomial inverse.
The Jacobian determinant is sometimes simply referred to as "the Jacobian". The Jacobian determinant at a given point gives important information about the behavior of f near that point. For instance, the continuously differentiable function f is invertible near a point p ∈ R n if the Jacobian determinant at p is non-zero.
For example, by the Jacobian criterion for regularity, a generic point of a variety over a field of characteristic zero is smooth. (This statement is known as generic smoothness .) This is true because the Jacobian criterion can be used to find equations for the points which are not smooth: They are exactly the points where the Jacobian matrix ...
Input: initial guess x (0) to the solution, (diagonal dominant) matrix A, right-hand side vector b, convergence criterion Output: solution when convergence is reached Comments: pseudocode based on the element-based formula above k = 0 while convergence not reached do for i := 1 step until n do σ = 0 for j := 1 step until n do if j ≠ i then ...
There is an analogous criterion for a continuously differentiable map f: R n → R n with a fixed point a, expressed in terms of its Jacobian matrix at a, J a (f). If all eigenvalues of J are real or complex numbers with absolute value strictly less than 1 then a is a stable fixed point; if at least one of them has absolute value strictly ...
Geometric regularity originated when Claude Chevalley and André Weil pointed out to Oscar Zariski that, over non-perfect fields, the Jacobian criterion for a simple point of an algebraic variety is not equivalent to the condition that the local ring is regular.
In mathematics, the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or function germ. Let (, ...
The Jacobian variety is named after Carl Gustav Jacobi, who proved the complete version of the Abel–Jacobi theorem, making the injectivity statement of Niels Abel into an isomorphism. It is a principally polarized abelian variety , of dimension g , and hence, over the complex numbers, it is a complex torus .