Three constructions of rational points onY 2=X3±NX

Distribution of Rational Points: A Survey

Rational Points

is known to have only finitely many triples of positive integer solutions x, y, z for a given n > 2 (Faltings, 1983). In Chapter 11, special situations are described in which more precise information is accessible. For example, if x is in S, then n is bounded by a computable number C5 = Cb(pv ..., p8). From these examples, it should be clear that the book is a mine of information for workers in...

Uniformity of Rational Points

1. Uniformity and correlation 1 1.

Rational Points on the Unit Sphere: Approximation Complexity and Practical Constructions

Each non-zero point in Rd identifies one closest point x on the unit sphere Sd−1. We are interested in computing an ε-approximation y ∈ Qd for x , that is exactly on Sd−1 and has low bit size. We revise lower bounds on rational approximations and provide explicit, spherical instances. We prove that floating-point numbers can only provide trivial solutions to the sphere equation in R2 and R3. Mo...

K3 Surfaces, Rational Curves, and Rational Points

We prove that for any of a wide class of elliptic surfaces X defined over a number field k, if there is an algebraic point on X that lies on only finitely many rational curves, then there is an algebraic point on X that lies on no rational curves. In particular, our theorem applies to a large class of elliptic K3 surfaces, which relates to a question posed by Bogomolov in 1981. Mathematics Subj...