Uniform bounds for curves and surfaces

Abstract

In 1989 Bombieri and Pila [7] pioneered a ground-breaking method for getting good upper bounds for the number of integer points on affine algebraic curves which are restricted to lie in a box. The most important feature of their estimate is its uniformity with respect to the particular curve. Thus if C⊂ \( \mathbb{A}^2 \) is an irreducible plane curve of degree d with integer coefficients, then they establish a bound of the form $$ \# (C(\mathbb{Z}) \cap [ - B,B]^2 ) = O_{d,\varepsilon } (B\tfrac{1} {d} + \varepsilon ), $$ (4.1) for any e>0. Here the implied constant depends at most upon the choice of e and the degree d. This is an extremely versatile result, which has already had a significant impact on many problems in analytic number theory. It also lies at the heart of an inductive proof (based on hyperplane sections) of the bound due to Pila [100], which appears as entry (iv) in Table 3.1. One notes that the Bombieri-Pila bound is essentially best possible, as consideration of the curve x d =y shows. One has ≫B1/a solutions of the form x=a, y=a d with modulus at most B.