Functiones et Approximatio Commentarii Mathematici

The integer points in a plane curve

Martin N. Huxley

Full-text: Open access

Abstract

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.

Article information

Source
Funct. Approx. Comment. Math., Volume 37, Number 1 (2007), 213-231.

Dates
First available in Project Euclid: 18 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.facm/1229618752

Digital Object Identifier
doi:10.7169/facm/1229618752

Mathematical Reviews number (MathSciNet)
MR2357320

Zentralblatt MATH identifier
1226.11106

Subjects
Primary: 11P21: Lattice points in specified regions
Secondary: 11J54: Small fractional parts of polynomials and generalizations 37C25: Fixed points, periodic points, fixed-point index theory

Keywords
determinant mean value theorem interpolation polynomial Grobner basis

Citation

Huxley, Martin N. The integer points in a plane curve. Funct. Approx. Comment. Math. 37 (2007), no. 1, 213--231. doi:10.7169/facm/1229618752. https://projecteuclid.org/euclid.facm/1229618752


Export citation

References

  • E. Bombieri and J. Pila, The number of integral points on arcs and ovals, Duke Math. J. 59 (1989), 337--357.
  • G.H. Hardy, A Course in Pure Mathematics ($10$th edn.), Cambridge University Press 1960.
  • M.N.Huxley, The rational points close to a curve, Ann. Scuola Norm. Sup. Pisa Cl. Sci. Fiz. Mat. (4) 21 (1994), 357--375.
  • M.N. Huxley, Area, Lattice Points, and Exponential Sums, London Math. Soc. Monographs 13, Oxford University Press 1996.
  • M.N. Huxley, The integer points close to a curve III, in Number Theory in Progress (ed. K. Györy et al.), (de Gruyter, Berlin 1999), vol II, 911--940.
  • M.N. Huxley, The rational points close to a curve II, Acta Arithmetica 93 (2000), 201--219.
  • M.N. Huxley, The rational points close to a curve III, Acta Arithmetica 113 (2004), 15--30.
  • M.N. Huxley, The rational points close to a curve IV, Bonner Math. Schriften, 360 (2003).
  • M.N. Huxley, Resonance curves in the Bombieri-Iwaniec method, Functiones et Approximatio 32 (2004), 7--49.
  • V. Jarník, Über die Gitterpunkte auf konvexen Kurven, Math. Zeitschrift 24 (1925), 500--518.
  • J. Pila, Geometric postulation of a smooth function and the number of rational points, Duke Math. J. 63 (1991), 449--463.
  • G. Pólya, On the mean-value theorem corresponding to a given linear homogeneous differential equation, Transactions Amer. Math. Soc. 24 (1922), 312--324.
  • W.M. Schmidt, Integer points on curves and surfaces, Monatshefte Math. 99 (1985), 45--72.
  • H.P.F. Swinnerton-Dyer, The number of lattice points on a convex curve, J. Number Theory 6 (1974), 128--135.
  • J.G. van der Corput, Over Roosterpunten in het Platte Vlaak, Nordhof, Groningen 1919.