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

TL;DR

This paper estimates the density of integer points on generalized Markoff-Hurwitz and Dwork hypersurfaces using bounds on mixed character sums, providing significantly improved results over previous methods.

Contribution

It introduces new bounds on mixed character sums to analyze the density of integer solutions on these hypersurfaces, extending and strengthening prior results.

Findings

Derived stronger bounds for integer point density

Applied bounds to specific hypersurfaces like Markoff-Hurwitz and Dwork

Achieved results surpassing previous estimates for general hypersurfaces

Abstract

We use bounds of mixed character sums modulo a square-free integer $q$ of a special structure 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 $f_i \in {\mathbb Z}[X]$ and nonzero integers $a$ and $k_i$, $i=1, \ldots, n$. 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 hypersurface 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 results are substantially stronger than those known for general hypersurfaces.