Enrico Bombieri

Minimal Cones and the Bernstein Problem.

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Canonical models of surfaces of general type

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On the large sieve

The purpose of this paper is to give a new and improved version of Linniks large sieve, with some applications. The large sieve has its roots in the Hardy-Littlewood method, and in its most general form it may be considered as an inequality which relates a singular series arising from an integral where S (α) is any exponential sum, to the integral itself.

The number of integral points on arcs and ovals

integral lattice points, and that the exponent and constant are best possible. However, Swinnerton–Dyer [10] showed that the preceding result can be substantially improved if we start with a fixed, C, strictly convex arc Γ and consider the number of lattice points on tΓ, the dilation of Γ by a factor t, t ≥ 1. This of course is the same as asking for rational points (mN , n N ) on Γ as N → ∞. In fact, Swinnerton–Dyer proves a bound of type |tΓ ∩ ZZ| ≤ c(Γ, e)t 3 5+e

Primes in arithmetic progressions to large moduli

Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Intersecting a curve with algebraic subgroups of multiplicative groups

Consider an arbitrary algebraic curve defined over the field of all alge- braic numbers and sitting in a multiplicative commutative algebraic group. In an earlier article from 1999 bearing almost the same title, we studied the intersection of the curve and the union of all algebraic subgroups of some fixed codimension. With codimension one the resulting set has bounded height properties, and with codimension two it has finiteness properties. The main aim of the present work is to make a start on such problems in higher dimension by proving the natural analogues for a linear surface (with codimensions two and three). These are in accordance with some general conjectures that we have recently proposed else- where.

On Thue's equation

and more precisely for the number of primitive solutions to (1.1), that is, solutions in coprime integers x, y. The first important results on the number of solutions of Thues equation were obtained by C.L. Siegel, in the case r = 3 and in the binomial case F ( x , y ) = a x r b y r. In view of his results and comments, we should attribute to Siegel the problem to decide whether the number of primitive solutions to (1.1), for irreducible F of degree r > 3, could be bounded by a function depending only on r and h, but otherwise independent of F. In 1983, Siegels question was answered in the affirmative by J.-H. Evertse [2]. As a special case of his results (he also treats equations in number fields), Evertse obtains the bound

Small Differences Between Prime Numbers

Let pn denote the nth prime number. The present investigation relates to the existence of relatively small values of pn+1─ pn when n is large, and establishes more precise results than those previously known.

Products of polynomials in many variables

Abstract We study the product of two polynomials in many variables, in several norms, and show that under suitable assumptions this product can be bounded from below independently of the number of variables.

On the Thue-Siegel-Dyson theorem

for q~qo(a, e). The constant qo(a, e) in this result turns out to be not effectively computable. In fact, Thue proved a result of this type with the exponent r/2+ l, Siegel improved this to the exponent min(r/(s+l)+s) for s=0, 1 . . . . . r 1 and finally, by using full freedom in the construction of the auxiliary polynomial, Dyson and Gelfond independently arrived at the exponent X/~. The common feature in the approach of Thue, Siegel, Dyson and Gelfond is the consideration of two approximations pl/ql,Pz]q2 to a and the construction of an auxiliary polynomial p(x~,x2) with integral coefficients vanishing to a high order at (a,a) and vanishing only to a low order at (Pt/qt,Pz/q2). Although Siegel and Schneider soon realized that further improvements could be obtained by the consideration of several distinct approximations P~/ql ..... Pm/qm to a and by the construction of an auxiliary polynomial P(x~, .... Xm) in many variables, it took about thirty years before Roth showed how to prove that P would vanish only to a low order at the point (P~/q~ ..... PMqm). In this way Roth was able to prove his celebrated theorem [ a---~ > q-2-e