• Corpus ID: 102341507

Rational points of bounded height on threefolds

  title={Rational points of bounded height on threefolds},
  author={Per Salberger},
Let ne,f (B) be the number of non-trivial positive integer solutions x0, x1, x2, y0, y1, y2 ≤ B to the simultaneous equations x0 + x e 1 + x e 2 = y e 0 + y e 1 + y e 2, x f 0 + x f 1 + x f 2 = y f 0 + y f 1 + y f 2 . We show that n1,4(B) = Oe(B), n1,5(B) = Oe(B) and that ne,f (B) = Oe,f,e(B 5/2+e) if ef ≥ 6 and f ≥ 4. These estimates are deduced from general upper bounds for the number of rational points of bounded height on projective threefolds over Q. 

Rational points on complete intersections of higher degree, and mean values of Weyl sums

We establish upper bounds for the number of rational points of bounded height on complete intersections. When the degree of the intersection is sufficiently large in terms of its dimension, and the

A paucity problem associated with a shifted integer analogue of the divisor function

Paucity problems and some relatives of Vinogradov's mean value theorem

When k > 4 and 0 6 d 6 (k− 2)/4, we consider the system of Diophantine equations x 1 + . . .+ xjk = y j 1 + . . .+ y k (1 6 j 6 k, j 6= k − d). We show that in this cousin of a Vinogradov system,

A paucity problem for certain triples of diagonal equations

We consider certain systems of three linked simultaneous diagonal equations in ten variables with total degree exceeding five. By means of a complification argument, we obtain an asymptotic formula


  • T. Wooley
  • Mathematics
    Bulletin of the Australian Mathematical Society
  • 2022
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Sur le nombre des entiers représentables comme somme de trois puissances

Dans cet article, nous établissons un résultat dans le cas des sommes de trois puissances de la forme suivante (1.4) n = c0n0 + c1n `1 1 + c2n `2 2 (n0, n1, n2 ∈ N), plus précisément des entiers



The density of rational points on non-singular hypersurfaces

AbstractLetF(x) =F[x1,…,xn]∈ℤ[x1,…,xn] be a non-singular form of degree d≥2, and letN(F, X)=#{xεℤn;F(x)=0, |x|⩽X}, where $$\left| x \right| = \mathop {max}\limits_{1 \leqslant r \leqslant n} \left|

Counting rational points on hypersurfaces

Abstract For any n ≧ 2, let F ∈ ℤ [ x 1, … , xn ] be a form of degree d≧ 2, which produces a geometrically irreducible hypersurface in ℙn–1. This paper is concerned with the number N(F;B) of rational

The density of rational points on curves and surfaces

Let $C$ be an irreducible projective curve of degree $d$ in $\mathbb{P}^3$, defined over $\overline{\mathbb{Q}}$. It is shown that $C$ has $O_{\varepsilon,d}(B^{2/d+\varepsilon})$ rational points of

Rational points of bounded height on projective surfaces

We give upper bounds for the number of rational points of bounded height on the complement of the lines on projective surfaces.

The paucity problem for simultaneous quadratic and biquadratic equations

  • W. TsuiT. Wooley
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1999
The problem of constructing non-diagonal solutions to systems of symmetric diagonal equations has attracted intense investigation for centuries (see [5, 6] for a history of such problems) and remains

Varieties with many lines

This paper contains a classification ofk-dimensional algebraic varieties,even non-smooth, containing a family of lines of dimension (2k−4). If the codimension of the varieties is greater than two,

Introduction to Algebraic Geometry

This classic work, now available in paperback, outlines the geometric aspects of algebraic equations, one of the oldest and most central subjects in mathematics. Recent decades have seen explosive

Automorphisms of Fermat-like varieties

Abstract. We study the automorphisms of some nice hypersurfaces and complete intersections in projective space by reducing the problem to the determination of the linear automorphisms of the ambient

Introduction to Grothendieck Duality Theory

Preface.- Study of ?X.- Completions, primary decomposition and length.- Depth and dimension.- Duality theorems.- Flat morphisms.- Etale morphisms.- Smooth morphisms.- Curves.