×

Integer points on hypersurfaces. (English) Zbl 0593.10015

It had been known that very general surfaces embedded in \(\mathbb{R}^3\) contain \(\ll r^{3/2}\) integer points in any ball of radius \(r>1\), and as a consequence, an algebraic hypersurface in \(\mathbb{R}^n\) \((n\ge 3)\) which is not a cylinder contains \(\ll r^{n-(3/2)}\) integer points in a ball of radius \(r\). Now it is shown that very general hypersurfaces in \(\mathbb{R}^4\) contain \(\ll r^{2+(4/9)}\) integer points in a ball of radius \(r>1\). As a consequence, an algebraic hypersurface in \(\mathbb{R}^n\) \((n\ge 4)\) which is not a cylinder contains \(\ll r^{n-1-(5/9)}\) integer points in such a ball.
The proof, which is based on the author’s earlier work, uses ideas from combinatorics and elementary geometry.

MSC:

11D72 Diophantine equations in many variables
11D41 Higher degree equations; Fermat’s equation
11H99 Geometry of numbers
PDFBibTeX XMLCite
Full Text: DOI EuDML

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] 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
[4] Heath-Brown, D. R.: Cubic forms in ten variables. Proc. Lond. Math. Soc. (3)47, 225–257 (1983). · Zbl 0509.10013 · doi:10.1112/plms/s3-47.2.225
[5] Schmidt, W. M.: Über Gitterpunkte auf gewissen Flächen. Mh. Math.68, 59–74 (1964). · Zbl 0131.29101 · doi:10.1007/BF01298826
[6] Schmidt, W. M.: Integer points on curves and surfaces. Mh. Math.99, 45–72 (1985). · Zbl 0551.10026 · doi:10.1007/BF01300739
[7] Swinnerton-Dyer, H. P. F.: The number of lattice points on a convex curve. J. Number Th.6, 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.