New sequences of non-free rational points

Distribution of Rational Points: A Survey

On Uniform Bounds for Rational Points on Non-rational Curves

We show that the number of rational points of height ≤ H on a non-rational plane curve of degree d is Od(H 2/d−δ), for some δ > 0 depending only on d. The implicit constant depends only on d. This improves a result of Heath-Brown, who proved the bound O (H2/d+ ). We also show that one can take δ = 1/450 in the case d = 3.

Elliptic Divisibility Sequences and Undecidable Problems about Rational Points

Julia Robinson has given a first-order definition of the rational integers Z in the rational numbers Q by a formula (∀∃∀∃)(F = 0) where the ∀-quantifiers run over a total of 8 variables, and where F is a polynomial. This implies that the Σ5-theory of Q is undecidable. We prove that a conjecture about elliptic curves provides an interpretation of Z in Q with quantifier complexity ∀∃, involving o...

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...

The density of rational points on non-singular hypersurfaces, I

For any n > 3, let F ∈ Z[X0, . . . , Xn] be a form of degree d > 5 that defines a non-singular hypersurface X ⊂ P. The main result in this paper is a proof of the fact that the number N(F ;B) of Q-rational points on X which have height at most B satisfies N(F ;B) = Od,ε,n(B ), for any ε > 0. The implied constant in this estimate depends at most upon d, ε and n. New estimates are also obtained f...