## Abstract

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}{k^{1}} 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) = aX^{2} and k_{1} k_{n} =1 the above congruence is known as the Markoff-Hurwitz hypersurface, while for f1(X)=fn(X) = X^{n} 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.

Original language | English |
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Pages (from-to) | 151-160 |

Number of pages | 10 |

Journal | International Journal of Number Theory |

Volume | 10 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 2014 |