On the density of rational lines on diagonal cubic hypersurfaces

TL;DR

The paper proves an asymptotic for the number of rational lines on diagonal cubic hypersurfaces for $s\\ge 19$, improving the threshold from the prior bound $s\\ge 21$. It combines the Hardy–Littlewood circle method with major/minor arc analysis, reducing the major-arc contribution to a product of a singular series and a singular integral, both bounded away from zero for these $s$. The minor-arc contribution is controlled by a multidimensional shifting-variables argument and a pruning technique that leverages sharp Parsell–Vinogradov bounds, leading to a precise mean-value bound and the final asymptotic $N_s(X)=\\sigma X^{2s-12}+O(X^{2s-12-\\delta})$. Overall, the work extends circle-method techniques to diagonal cubic forms and lays groundwork for further extensions to higher degrees and more general 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 $s\geq 19.$ 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.

Theorems & Definitions (11)

• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 3.1
• proof
• Lemma 4.1
• proof
• Lemma 4.2
• proof