Open Access
January 2007 The integer points in a plane curve
Martin N. Huxley
Funct. Approx. Comment. Math. 37(1): 213-231 (January 2007). DOI: 10.7169/facm/1229618752


Bombieri and Pila gave sharp estimates for the number of integer points $(m,n)$ on a given arc of a curve $y = F(x)$, enlarged by some size parameter $M$, for algebraic curves and for transcendental analytic curves. The transcendental case involves the maximum number of intersections of the given arc by algebraic curves of bounded degree. We obtain an analogous result for functions $F(x)$ of some class $C^k$ that satisfy certain differential inequalities that control the intersection number. We allow enlargement by different size parameters $M$ and $N$ in the $x$- and $y$-directions, and we also estimate integer points close to the curve, with $$\left|n - NF ( {m\over M} )| \leq \delta,$$ for $\delta$ sufficiently small in terms of $M$ and $N$. As an appendix we obtain a determinant mean value theorem which is a quantitative version of a linear independence theorem of Pólya.


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Martin N. Huxley. "The integer points in a plane curve." Funct. Approx. Comment. Math. 37 (1) 213 - 231, January 2007.


Published: January 2007
First available in Project Euclid: 18 December 2008

zbMATH: 1226.11106
MathSciNet: MR2357320
Digital Object Identifier: 10.7169/facm/1229618752

Primary: 11P21
Secondary: 11J54 , 37C25

Keywords: determinant mean value theorem , Grobner basis , interpolation , polynomial

Rights: Copyright © 2007 Adam Mickiewicz University

Vol.37 • No. 1 • January 2007
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