On the density of rational lines on diagonal cubic hypersurfaces

TL;DR

This paper improves the understanding of the distribution of rational lines on diagonal cubic hypersurfaces by establishing new asymptotic estimates for cases with at least 19 variables, using advanced analytic techniques.

Contribution

It reduces the variable bound needed for asymptotic estimates from 21 to 19 by developing a novel multidimensional shifting and pruning method.

Findings

Established asymptotic estimates for rational lines with s ≥ 19

Improved previous bounds from s ≥ 21 to s ≥ 19

Developed new analytic techniques involving multidimensional shifting and pruning

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 $s\geq 19.$ This improves the previously known bound $s\geq 21$ required to obtain such asymptotic estimates. Our approach develops a multidimensional shifting variables argument together with a pruning argument, and exploits the recent progress on the Parsell-Vinogradov system.