## 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
Full Text:

### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.