Consider an elliptic curve, defined over the rational numbers, and embedded in projective space. The rational points on the curve are viewed as integer vectors with coprime coordinates. What can be said about the rational points for which the number of prime factors dividing a fixed coordinate does not exceed a fixed bound? If the bound is zero, then Siegel’s Theorem guarantees that there are o...

We answer a question asked by Hajdu and Tengely: The only arithmetic progression in coprime integers of the form (a, b, c, d) is (1, 1, 1, 1). For the proof, we first reduce the problem to that of determining the sets of rational points on three specific hyperelliptic curves of genus 4. A 2-cover descent computation shows that there are no rational points on two of these curves. We find generat...

Consider a rational point on an elliptic curve under an isogeny. Suppose that the action of Galois partitions the set of its preimages into n orbits. It is shown that all but finitely many such points have their denominator divisible by at least n distinct primes. This generalizes Siegel’s theorem and more recent results of Everest et al. For multiplication by a prime l, it is shown that if n >...

1. Rational points. A classical conjecture of Mordell states that a curve of genus ^ 2 over the rational numbers has only a finite number of rational points. Let K be a finitely generated field over the rational numbers. Then the same statement should hold for a curve defined over K, and a specialization argument due to Néron shows in fact that this latter statement is implied by the correspond...

A piecewise rational cubic spline [5] has been used to visualize the positive data in its natural form. The spline representation is interpolatory and applicable to the scalar valued data. The shape parameters in the description of a rational cubic have been constrained in such a way that they preserve the shape of the positive data in the view of positive curve. As far as visual smoothness is ...

In [14], D. Skabelund constructed a maximal curve over Fq4 as cyclic cover of the Suzuki curve. this paper we explicitly determine structure Weierstrass semigroup at any point P We show that its points are precisely Fq4-rational points. Also among points, two types occur: one for Fq-rational remaining For each these Apéry set is computed well generators.

Given a curve C on a projective nonsingular rational surface S, over an algebraically closed field of characteristic zero, we are interested in the set ΩC of linear systems L on S satisfying C ∈ L, dimL > 1, and the general member of L is a rational curve. The main result of the paper gives a complete description of ΩC and, in particular, characterizes the curves C for which ΩC is non empty. 20...

Based on the technique of C-shape G Hermite interpolation by cubic Pythagorean-Hodograph (PH) curve, we present a simple method for Cshape G Hermite interpolation by rational cubic Bézier curve. The method reproduces a circular arc when the input data come from it. Both the Bézier control points, which have the well understood geometrical meaning, and the weights of the resulting rational cubic...

(2) hn divides hm whenever n divides m. They have attracted number theoretical and combinatorial interest as some of the simplest nonlinear recurrence sequences (see [3] for references), but for us their interest lives in the underlying geometry: Ward demonstrates that an elliptic divisibility sequence arises from any choice of elliptic curve over Q and rational point on that curve. Theorem 1 (...

We present a fast addition algorithm in the Jacobian of a genus 3 non-hyperelliptic curve over a field of any characteristic. When the curve has a rational flex and char(k) > 5, the computational cost for addition is 148M + 15SQ + 2I and 165M + 20SQ + 2I for doubling. An appendix focuses on the computation of flexes in all characteristics. For large odd q, we also show that the set of rational ...