On the distribution of points on the generalized markoff-hurwitz and dwork hypersurfaces

We use bounds of mixed character sum modulo a prime p to study the distribution of points on the hypersurface f1(x1) fn(xn) ≡ x ₁{k¹} x_n ^kn(mod p)for some polynomials f_i [X] that are not constant modulo a prime p and integers k_i with gcd(k_i, p-1) = 1, i = 1, n. In the case of f1(X) fn(X) = aX² and k₁ k_n =1 the above congruence is known as the Markoff-Hurwitz hypersurface, while for f1(X)=fn(X) = Xⁿ and k1=kn =1 it is known as the Dwork hypersurface. In particular, we obtain non-trivial results about the number of solution in boxes with the side length below p^1/2, which seems to be the limit of more general methods based on the bounds of exponential sums along varieties.