The density of rational lines on hypersurfaces: A bihomogeneous perspective

TL;DR

This paper derives an asymptotic formula for counting pairs of integer points that generate lines on a hypersurface defined by a non-singular homogeneous polynomial, extending previous results by removing size restrictions.

Contribution

It provides a new asymptotic count of rational lines on hypersurfaces using a bihomogeneous perspective, without size restrictions on the points.

Findings

Asymptotic formula for pairs of integer points generating lines in hypersurfaces

Applicable to non-singular homogeneous polynomials of degree d

No restrictions on relative sizes of X and Y in the count

Abstract

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 $Y$.