English

On the density of rational lines on diagonal cubic hypersurfaces

Number Theory 2026-02-05 v1

Abstract

In this paper, we establish the asymptotic estimates for the rational lines on diagonal cubic hypersurfaces defined by i=1scixi3=0\sum_{i=1}^sc_ix^3_i=0 with ciZ{0},c_i\in\mathbb{Z}\setminus \{0\}, provided that s19.s\geq 19. This improves the previously known bound s21s\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.

Keywords

Cite

@article{arxiv.2602.04654,
  title  = {On the density of rational lines on diagonal cubic hypersurfaces},
  author = {Kiseok Yeon},
  journal= {arXiv preprint arXiv:2602.04654},
  year   = {2026}
}

Comments

14 pages, all comments welcome!