On the Density of Integer Points on Generalised Markoff-Hurwitz and Dwork 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 $$f_1(x_1)+\dots +f_n(x_n)=a{x}_1^{k_1}\dots x_n^{k_n}$$ for some polynomials $f_i\in \mathbb{Z}[X]$ and nonzero integers $a$ and $k_i$, $i=1,\dots ,n$. In the case of $$f_1(X)=\dots =f_n(X)=X^2\text{and}{k}_1=\dots =k_n=1$$ the above hypersurface is known as the Markoff-Hurwitz hypersurface, while for $$f_1(X)=\dots =f_n(X)=X^n\text{and}{k}_1=\dots =k_n=1$$ it is known as the Dwork hypersurface. Our results are substantially stronger than those known for general hypersurfaces.