Throughout this paper, k will denote a commutative ring containing the rational numbers Q, and k = k[x{, . . . , xn] will be the polynomial ring over k . If ƒ : k —• k is a polynomial map (i.e., a fc-algebra homomorphism), then ƒ is a polynomial automorphism provided there is an inverse ƒ " which is also a polynomial map. Very little is known about the group of polynomial automorphisms, and ind...

We have two main results. 1. Let P : Kn → Kn be a polynomial map with constant nonzero Jacobian, where K is any extension of Q. Then, P has a polynomial inverse if and only if the range of P contains a cartesian product of n universal Hilbert sets. 2. Let P : K → K be a polynomial map with constant nonzero Jacobian, where K is an algebraic number field. Then, P is invertible for “almost all” ra...

Introduction The main object of this paper is to prove the following result: Theorem 1 The Jacobian of a real analytic open map f : R → R does not change sign. One of the referees kindly pointed out that the special case of polynomial maps was proved by Gamboa and Ronga [3]: Theorem 2 (Gamboa and Ronga) A polynomial map in R is open if and only if point inverses are finite and the Jacobian does...

The hyperelliptic curve cryptosystems take most of the time for computing a scalar multiplication kD of an element D in the Jacobian JC of a hyperelliptic curve C for an integer k. Therefore its efficiency depends on the scalar multiplications. Among the fast scalar multiplication methods, there is a method using a Frobenius map. It uses a Jacobian defined over an extension field of the definit...

The distributional k-dimensional Jacobian of a map u in the Sobolev space W 1,k−1 which takes values in the the sphere S can be viewed as the boundary of a rectifiable current of codimension k carried by (part of) the singularity of u which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian operator; in particular, we show that any boundary M o...

We give a full proof of the two dimensional Jacobian conjecture. We also give an algorithm to compute the inverse map of a polynomial map.

According to [LO], the Prym variety of any non-cyclic étale triple cover f : Y → X of a smooth curve X of genus 2 is a Jacobian variety of dimension 2. This gives a map from the moduli space of such covers to the moduli space of Jacobian varieties of dimension 2. We extend this map to a proper map Pr of a moduli space S3M̃2 of admissible S3-covers of genus 7 to the moduli space A2 of principally...

a watermarking scheme for Grayscale image isproposed based on a family of the chaotic maps and discretecosine transform. Jacobian Elliptic mapis employed to encrypt ofwatermarked logo. Piecewise nonlinear chaotic map is also usedto determine the location of DCT coefficients for the watermarkembedding. The purpose of this algorithm is to improve theshortcoming of watermarking such as small key s...

For redundant robot kinematics we construct an extended Jacobian inverse that approximates the Jacobian pseudo-inverse. Our construction relies on the approximation of codistributions associated with the Jacobian pseudo-inverse by an integrable codistribution, spanned by differentials of certain functions that serve as components of the augmenting kinematics map in the extended Jacobian inverse...