Let F be a non-singular homogeneous polynomial of degree d in n variables. We give an asymptotic formula of the pairs of integer points \((\mathbf {x}, \mathbf {y})\) with \(|\mathbf {x}| \leqslant X\) and \(|\mathbf {y}| \leqslant Y\) which generate a line lying in the hypersurface defined by F, provided that \(n > 2^{d-1}d^4(d+1)(d+2)\). In particular, by restricting to Zariski-open subsets we are able to avoid imposing any conditions on the relative sizes of X and Y.

中文翻译：

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超曲面上有理线的密度：双同质透视

让˚F是程度的非奇异均匀多项式d在ñ变量。我们给出整数点对\（（\ mathbf {x}，\ mathbf {y}）\）与\（| \ mathbf {x} | \ leqslant X \）和\（| \ mathbf { y} | \ leqslant Y \）生成位于F定义的超曲面中的一条线，条件是\（n> 2 ^ {d-1} d ^ 4（d + 1）（d + 2）\）。特别地，通过限制Zariski-open子集，我们可以避免对X和Y的相对大小施加任何条件。