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.


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: DOI EuDML


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