problem: Given: A field K and n variables x1, . . . , xn and m polynomials yi = pi(x1, . . . , xn) ∈ K[x1, . . . , xn] for i = 1, . . . ,m. (1) Aim: Find a presentation for the subring K[y] := K[y1, . . . , ym] of K[x] := K[x1, . . . , xn]. Invariants: The difference of m and the transcendence degree of K(y) := K(y1, . . . , ym) over K will be called the deficiency d = d(y) of the tuple y in K(...

We construct Weil numbers corresponding to genus-2 curves with p-rank 1 over the finite field Fp2 of p 2 elements. The corresponding curves can be constructed using explicit CM constructions. In one of our algorithms, the group of Fp2 -valued points of the Jacobian has prime order, while another allows for a prescribed embedding degree with respect to a subgroup of prescribed order. The curves ...

We present several explicit constructions of hyperelliptic function fields whose Jacobian or ideal class group has large 3-rank. Our focus is on finding examples for which the genus and the base field are as small as possible. Most of our methods are adapted from analogous techniques used for generating quadratic number fields whose ideal class groups have high 3-rank, but one method, applicabl...

We construct examples of families of curves of genus 2 or 3 over Q whose Jacobians split completely and have various large rational torsion subgroups. For example, the rational points on a certain elliptic surface over P of positive rank parameterize a family of genus-2 curves over Q whose Jacobians each have 128 rational torsion points. Also, we find the genus-3 curve 15625(X + Y 4 + Z)− 96914...

A technique of descent via 4-isogeny is developed on the Jacobian of a curve of genus 2 of the form: Y 2 = q1(X)q2(X)q3(X), where each qi(X) is a quadratic defined over Q. The technique offers a realistic prospect of calculating rank tables of Mordell-Weil groups in higher dimension. A selection of worked examples is included as illustration.

We consider Newton’s algorithm as well as a variant, the Gauss– Newton algorithm, to solve a system of nonlinear equations F (x) = 0, where x ∈ R , F : R → R. We use a line search method to ensure global convergence. The exact form of our algorithm depends on the rank of the Jacobian J(x) of F . Computational results on some standard test problems are presented, which show that the algorithm ma...

We construct examples of families of curves of genus 2 or 3 over Q whose Jacobians split completely and have various large rational torsion subgroups. For example, the rational points on a certain elliptic surface over P of positive rank parameterize a family of genus-2 curves over Q whose Jacobians each have 128 rational torsion points. Also, we find the genus-3 curve 15625(X4 + Y 4 + Z4)− 969...

This paper presents an iterative scheme for solving nonline ar equations. We establish a new rational approximation model with linear numerator and denominator which has generalizes the local linear model. We then employ the new approximation for nonlinear equations and propose an improved Newton's method to solve it. The new method revises the Jacobian matrix by a rank one matrix each iteratio...

Recent developments in formulations for generating swept volumes have made a significant impact on the efficiency of employing such algorithms and on the extent to which formulations can be used in representing complex shapes. In this paper, we outline a method for employing the representation of implicit surfaces using the Jacobian rank deficiency condition presented earlier for the sweep of p...