Abstract
Various upper bounds are given for the number of integer points on plane curves, on surfaces and hypersurfaces. We begin with a certain class of convex curves, we treat rather general surfaces in ℝ3 which include algebraic surfaces with the exception of cylinders, and we go on to hypersurfaces in ℝn with nonvanishing Gaussian curvature.
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Andrews, G. E.: An asymptotic expression for the number of solutions of a general class of diophantine equations. Trans. Amer. Math. Soc.99, 272–277 (1961).
Andrews, G. E.: A lower bound for the volume of strictly convex bodies with many boundary lattice points. Trans. Amer. Math. Soc.106, 270–279 (1963).
Cassels, J. W. S.: An Introduction to the Geometry of Numbers. Grundlehren 99. Berlin-Heidelberg-New York: Springer. 1959.
Cohen, S. D.: The distribution of Galois groups and Hilbert's irreducibility theorem. Proc. Lond. Math. Soc. (3)43, 227–250 (1981).
Davenport, H.: Indefinite quadratic forms in many variables (II). Proc. Lond. Math. Soc. (3)8, 109–126 (1958).
Grünbaum, B.: Convex Polytopes. Interscience Publ. 1967.
Heath-Brown, D. R.: Cubic forms in ten variables. Proc. Lond. Math. Soc. (3)47, 225–257 (1983).
Jarnik, V.: Über die Gitterpunkte auf konvexen Kurven. Math. Z.24, 500–518 (1925).
Roth, K. F. Rational approximations to algebraic numbers. Mathematika2, 1–20 (1955).
Schmidt, W. M.: Über Gitterpunkte auf gewissen Flächen. Mh. Math.68, 59–74 (1964).
Swinnerton-Dyer, H. P. F.: The number of lattice points on a convex curve. J. Number Theory6, 128–135 (1974).
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Written with partial supports from NSF grant No. MCS-8211461.
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Schmidt, W.M. Integer points on curves and surfaces. Monatshefte für Mathematik 99, 45–72 (1985). https://doi.org/10.1007/BF01300739
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DOI: https://doi.org/10.1007/BF01300739