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Forms in many variables and differing degrees
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, weakExpand
Power-free values of polynomials
For an irreducible polynomial in at most two variables the problem of representing power-free integers is investigated.
Counting Rational Points on Algebraic Varieties
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 " > 0Expand
Improvements in Birch's theorem on forms in many variables
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 theExpand
Quantitative Arithmetic of Projective Varieties
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 andExpand
Sums of arithmetic functions over values of binary forms
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, inExpand
Analytic Methods for Diophantine Equations and Diophantine Inequalities: Introduction
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. TheExpand
Rational points on quartic hypersurfaces
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-adicExpand