论文标题
Hypersurfaces上理性线的密度：双重观点
The density of rational lines on hypersurfaces: A bihomogeneous perspective
论文作者
论文摘要
令$ f $为$ n $变量的非单明性均值$ d $。我们将$ | \ mathbf x |的整数点$（\ mathbf x，\ mathbf y）$提供渐近公式。 \ le x $和$ | \ Mathbf y |如果$ n> 2^{d-1} d^4（d+1）（d+2）$，则在$ f $定义的高表面上产生一条线。特别是，通过限制Zariski-open子集，我们可以避免对$ x $和$ y $的相对大小强加任何条件。
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 $|\mathbf x| \le X$ and $|\mathbf y| \le Y$ 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 $Y$.
