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 f(1) (x(1)) + . . . + f(n) (x(n)) x(1)(k1) . . . x(n)(kn) (mod p) for some polynomials f(i) is an element of Z[X] that are not constant modulo a prime p and integers ki with gcd (k(i), p - 1) = 1, i = 1, . . . , n. In the case of f(1)(X) = . . . = f(n) (X) = aX(2) and k(1) = . . . = k(n) = 1 the above congruence is known as the Markoff-Hurwitz hypersurface, while for f(1)(X) = . . . = f(n) (X) = X-n and k(1) = . . . = k(n) = 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.