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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 $$\begin{aligned} f_1(x_1) + \cdots + f_n(x_n) =a x_1^{k_1} \ldots x_n^{k_n} \end{aligned}$$for some polynomials \(f_i \in {\mathbb {Z}}[X]\) and nonzero integers a and \(k_i\), \(i=1, \ldots , n\). In the case of $$\begin{aligned} f_1(X) = \cdots = f_n(X) = X^2\quad \text{ and }\quad k_1 = \cdots = k_n =1 \end{aligned}$$the above hypersurface is known as the Markoff–Hurwitz hypersurface, while for $$\begin{aligned} f_1(X) = \cdots = f_n(X) = X^n\quad \text{ and }\quad k_1 = \cdots = k_n =1 \end{aligned}$$it is known as the Dwork hypersurface. Our results are substantially stronger than those known for general hypersurfaces. Keywords Integer points on hypersurfaces Multiplicative character sums Congruences Mathematics Subject Classification 11D45 11D72 11L40 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (30) References1.Baragar, A.: Asymptotic growth of Markoff–Hurwitz numbers. Compos. Math. 94, 1–18 (1994)MathSciNetMATH2.Baragar, A.: The exponent for the Markoff–Hurwitz equations. Pac. J. Math. 182, 1–21 (1998)MathSciNetCrossRefMATH3.Baragar, A.: The Markoff–Hurwitz equations over number fields. Rocky Mt. J. Math. 35, 695–712 (2005)MathSciNetCrossRefMATH4.Birch, B.J.: Forms in many variables. Proc. R. Soc. Ser. 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