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Integer points on curves and surfaces. (English) Zbl 0551.10026

Let \({\mathfrak C}\) be a curve \(y=f(x)\) where x runs through an interval of length \(N\geq 1\), and suppose that f” exists and is increasing. By using a variation on a method of Swinnerton-Dyer it is shown that the number of integer points on \({\mathfrak C}\) is \(\leq c_ 1(\epsilon)N^{(3/5)+\epsilon}.\)
Let \({\mathfrak S}\) be a surface in \({\mathbb{R}}^ 3\). By using earlier work of the author it is shown that under mild assumptions on \({\mathfrak S}\), the number of integer points on \({\mathfrak S}\) in a cube of side \(N\geq 1\) is \(\leq c_ 2N^{3/2}\). It follows that when \({\mathfrak A}\) is an irreducible algebraic hypersurface of degree d in \({\mathbb{R}}^ n\) (n\(\geq 3)\) which is not a cylinder, then the number of integer points on S in a cube of side \(N\geq 1\) is \(\leq c_ 3(n,d)N^{n-(3/2)}\). A new proof is given of a theorem of Andrews on integer points on convex surfaces. Finally, given \(1\leq r<n\), an upper bound is obtained concerning integer points on hypersurfaces in \({\mathbb{R}}^ n\) having at least r positive curvatures everywhere.

MSC:

11H99 Geometry of numbers
11H06 Lattices and convex bodies (number-theoretic aspects)
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
11D61 Exponential Diophantine equations
11P21 Lattice points in specified regions
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References:

[1] 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). · Zbl 0113.03702 · doi:10.1090/S0002-9947-1961-0120222-7
[2] 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). · Zbl 0118.28301 · doi:10.1090/S0002-9947-1963-0143105-7
[3] Cassels, J. W. S.: An Introduction to the Geometry of Numbers. Grundlehren 99. Berlin-Heidelberg-New York: Springer. 1959. · Zbl 0086.26203
[4] Cohen, S. D.: The distribution of Galois groups and Hilbert’s irreducibility theorem. Proc. Lond. Math. Soc. (3)43, 227-250 (1981). · Zbl 0484.12002 · doi:10.1112/plms/s3-43.2.227
[5] Davenport, H.: Indefinite quadratic forms in many variables (II). Proc. Lond. Math. Soc. (3)8, 109-126 (1958). · Zbl 0078.03901 · doi:10.1112/plms/s3-8.1.109
[6] Grünbaum, B.: Convex Polytopes. Interscience Publ. 1967.
[7] Heath-Brown, D. R.: Cubic forms in ten variables. Proc. Lond. Math. Soc. (3)47, 225-257 (1983). · doi:10.1112/plms/s3-47.2.225
[8] Jarnik, V.: Über die Gitterpunkte auf konvexen Kurven. Math. Z.24, 500-518 (1925). · doi:10.1007/BF01216795
[9] Roth, K. F. Rational approximations to algebraic numbers. Mathematika2, 1-20 (1955). · Zbl 0064.28501 · doi:10.1112/S0025579300000644
[10] Schmidt, W. M.: Über Gitterpunkte auf gewissen Flächen. Mh. Math.68, 59-74 (1964). · Zbl 0131.29101 · doi:10.1007/BF01298826
[11] Swinnerton-Dyer, H. P. F.: The number of lattice points on a convex curve. J. Number Theory6, 128-135 (1974). · Zbl 0285.10020 · doi:10.1016/0022-314X(74)90051-1
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