### 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

for some polynomials

the above hypersurface is known as the Markoff–Hurwitz hypersurface, while for

*ƒ*_{1}(x_{1}) +···+*ƒ*_{n}(x_{n}) =*ax*_{1}^{k1 }...*x*_{n}^{kn}for some polynomials

*ƒ*_{i}∈ Z[*X*] and nonzero integers*a*and*k*_{i}, i = 1,...,n. In the case of*ƒ*_{1}(*X*) = ··· =*ƒ*_{n}(*X*) =*X*^{2}and k_{1}=··· = k_{n}= 1the above hypersurface is known as the Markoff–Hurwitz hypersurface, while for

*ƒ*_{1}(*X*) = ··· =*ƒ*_{n}(*X*) =*X*^{n}and k_{1}=··· = k_{n}= 1 it is known as the Dwork hypersurface. Our results are substantially stronger than those known for general hypersurfaces.Original language | English |
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Pages (from-to) | 935-954 |

Number of pages | 20 |

Journal | Mathematische Zeitschrift |

Volume | 282 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - Apr 2016 |

Externally published | Yes |

### Keywords

- Congruences
- Integer points on hypersurfaces
- Multiplicative character sums

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## Cite this

Chang, M. C., & Shparlinski, I. E. (2016). On the density of integer points on generalised Markoff–Hurwitz and Dwork hypersurfaces.

*Mathematische Zeitschrift*,*282*(3-4), 935-954. https://doi.org/10.1007/s00209-015-1571-z