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arXiv · 2405.04234

On a conjecture of Wooley and lower bounds for cubic hypersurfaces

V. Vinay Kumaraswamy·Nick Rome
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Abstract

Let $X \subset \mathbf{P}_{\mathbf{Q}}^{n-1}$ be a cubic hypersurface cut out by the vanishing of a non-degenerate rational cubic form in $n$ variables. Let $N(X,B)$ denote the number of rational points on $X$ of height at most $B$. In this article we obtain lower bounds for $N(X,B)$ for cubic hypersufaces, provided only that $n$ is large enough. In particular, we show that $N(X,B) \gg B^{n-9}$ if $n \geq 39$, thereby proving a conjecture of T. D. Wooley for non-conical cubic hypersurfaces with large enough dimension.

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V. Vinay Kumaraswamy, Nick Rome. 2024-05-07. On a conjecture of Wooley and lower bounds for cubic hypersurfaces. https://arxiv.org/abs/2405.04234

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