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

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.