On the density of rational lines on cubic diagonal hypersurfaces

Abstract: In this paper, we establish the asymptotic estimates for the rational lines on diagonal cubic hypersurfaces defined by $\sum_{i=1}^sc_ix^3_i=0$ with $c_i\in\mathbb{Z}\setminus \{0\},$ provided that $s\geq 18.$ This improves the previously known bound $s\geq 21$ required to obtain such asymptotic estimates. Our approach develops a multidimensional shifting variables argument, and exploits the recent progress on the Parsell-Vinogradov system. This talk is based on the speaker’s recent work and a joint work with Parsell.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)