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Forms in many variables and differing degrees

- T. Browning, D. R. Heath-Brown
- Mathematics
- 24 March 2014

We generalise Birch's seminal work on forms in many variables to handle a system of forms in which the degrees need not all be the same. This allows us to prove the Hasse principle, weak… Expand

Power-free values of polynomials

- T. Browning
- Mathematics
- 19 February 2011

For an irreducible polynomial in at most two variables the problem of representing power-free integers is investigated.

Counting Rational Points on Algebraic Varieties

- T. Browning, D. R. Heath-Brown, P. Salberger
- Mathematics
- 5 October 2004

For any N � 2, let ZP N be a geometrically integral algebraic variety of degree d. This paper is concerned with the number NZ(B) of Q-rational points on Z which have height at most B. For any " > 0… Expand

Analytic Methods for Diophantine Equations and Diophantine Inequalities: Cubic forms: bilinear equations

- H. Davenport, T. Browning
- Mathematics
- 2005

Improvements in Birch's theorem on forms in many variables

- T. Browning, Sean Prendiville
- Mathematics
- 18 February 2014

We show that a non-singular integral form of degree d is soluble non-trivially over the integers if and only if it is soluble non-trivially over the reals and the p-adic numbers, provided that the… Expand

Quantitative Arithmetic of Projective Varieties

- T. Browning
- Mathematics
- 23 October 2009

The Manin conjectures.- The dimension growth conjecture.- Uniform bounds for curves and surfaces.- A1 del Pezzo surface of degree 6.- D4 del Pezzo surface of degree 3.- Siegel's lemma and… Expand

Sums of arithmetic functions over values of binary forms

- R. Breteche, T. Browning
- Mathematics
- 5 April 2006

Given a suitable arithmetic function h, we investigate the average order of h as it ranges over the values taken by an integral binary form F. A general upper bound is obtained for this quantity, in… Expand

Analytic Methods for Diophantine Equations and Diophantine Inequalities: Introduction

- H. Davenport, T. Browning
- Mathematics
- 1962

Preface Foreword 1. Introduction 2. Waring's problem: history 3. Weyl's inequality and Hua's inequality 4. Waring's problem: the asymptotic formula 5. Waring's problem: the singular series 6. The… Expand

Rational points on quartic hypersurfaces

- T. Browning, D. R. Heath-Brown
- Mathematics
- 12 January 2007

Abstract Let X be a projective non-singular quartic hypersurface of dimension 39 or more, which is defined over ℚ. We show that X(ℚ) is non-empty provided that X(ℝ) is non-empty and X has p-adic… Expand

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