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The integer points on a plane curve

Huxley, Martin Neil 2007. The integer points on a plane curve. Functiones et Approximatio, Commentarii Mathematici 37 (1) , pp. 213-231.

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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.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Publisher: Adam Mickiewicz University
ISSN: 0208-6573
Last Modified: 04 Jun 2017 04:01

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