On the Density of Integer Points on the Generalised Markoff-Hurwitz and Dwork Hypersurfaces

TL;DR

This paper estimates the density of integer points on certain polynomial hypersurfaces using bounds on mixed character sums, providing significantly improved results for Markoff-Hurwitz and Dwork hypersurfaces.

Contribution

It introduces new bounds for integer point density on polynomial hypersurfaces, especially strengthening results for Markoff-Hurwitz and Dwork cases.

Findings

Stronger bounds on integer point density for these hypersurfaces.

Application of mixed character sum bounds to hypersurface point counting.

Improved estimates over previous general hypersurface results.

Abstract

We use bounds of mixed character sums modulo a prime $p$ to estimate the density of integer points on the hypersurface $$ f_1(x_1) + \ldots + f_n(x_n) =a x_1^{k_1} \ldots x_n^{k_n} $$ for some polynomials $f_i \in {\mathbb Z}[X]$, nonzero integer $a$ and positive integers $k_i$ $i=1, \ldots, n$. In the case of $$ f_1(X) = \ldots = f_n(X) = X^2 \quad \text{and}\quad k_1 = \ldots = k_n =1 $$ the above congruence is known as the Markoff-Hurwitz hypersurface, while for $$ f_1(X) = \ldots = f_n(X) = X^n\quad \text{and}\quad k_1 = \ldots = k_n =1 $$ it is known as the Dwork hypersurface. Our result is substantially stronger than those known for general hypersurfaces.