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[8] [9] Some authors define the Jacobian as the transpose of the form given above. The Jacobian matrix represents the differential of f at every point where f is differentiable. In detail, if h is a displacement vector represented by a column matrix , the matrix product J ( x ) ⋅ h is another displacement vector, that is the best linear ...
For a given F, the Jacobian conjecture is true if, and only if, k[X] = k[F]. Keller (1939) proved the birational case, that is, where the two fields k(X) and k(F) are equal. The case where k(X) is a Galois extension of k(F) was proved by Andrew Campbell for complex maps [8] and in general by Michael Razar [9] and, independently, by David Wright ...
More generally, if the determinant of A is positive, A represents an orientation-preserving linear transformation (if A is an orthogonal 2 × 2 or 3 × 3 matrix, this is a rotation), while if it is negative, A switches the orientation of the basis.
For m = 0 the generalized Jacobian J m is just the usual Jacobian J, an abelian variety of dimension g, the genus of C. For m a nonzero effective divisor the generalized Jacobian is an extension of J by a connected commutative affine algebraic group L m of dimension deg(m)−1. So we have an exact sequence 0 → L m → J m → J → 0
The Jacobian of a curve over an arbitrary field was constructed by Weil (1948) as part of his proof of the Riemann hypothesis for curves over a finite field. The Abel–Jacobi theorem states that the torus thus built is a variety, the classical Jacobian of a curve, that indeed parametrizes the degree 0 line bundles, that is, it can be ...
An elliptic curve E, over a field K can be put in the Weierstrass form y 2 = x 3 + ax + b, with a, b in K. What will be of importance later are point of order 2, that is P on E such that [2]P = O and P ≠ O. If P = (p, 0) is a point on E, then it has order 2; more generally the points of order 2 correspond to the roots of the polynomial f(x ...
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities.
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [1]If A is a differentiable map from the real numbers to n × n matrices, then