Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Integral Points on Elliptic Curves and the Bombieri-Pila Bounds   Let C be an affine, plane, algebraic curve of degree d with integer
coefficients. In 1989, Bombieri and Pila showed that if one takes a box with
sides of length N then C can obtain no more than
O_{d,\epsilon}(N^{1/d+\epsilon}) integer points within the box. Importantly,
the implied constant makes no reference to the coefficients of the curve.
Examples of certain rational curves show that this bound is tight but it has
long been thought that when restricted to non-rational curves an improvement
should be possible whilst maintaining the uniformity of the bound. In this
paper we consider this problem restricted to elliptic curves and show that for
a large family of these curves the Bombieri-Pila bounds can be improved. The
techniques involved include repulsion of integer points, the theory of heights
and the large sieve. As an application we prove a uniform bound for the number
of rational points of bounded height on a general del Pezzo surface of degree
1.

23
The categories of this paper are: solv-int
30
The categories of this paper are: q-alg
8
The categories of this paper are: hep-ex
17
The categories of this paper are: q-fin
22
The categories of this paper are: chao-dyn
28
The categories of this paper are: alg-geom
5
The categories of this paper are: gr-qc
0
The categories of this paper are: hep-ph
3
The categories of this paper are: physics
20
The categories of this paper are: acc-phys
25
The categories of this paper are: patt-sol
9
The categories of this paper are: nlin
10
The categories of this paper are: q-bio
19
The categories of this paper are: eess
15
The categories of this paper are: nucl-ex
24
The categories of this paper are: dg-ga