On a conjecture of Wooley and lower bounds for cubic hypersurfaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916238712635392 |
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| author | Kumaraswamy, V. Vinay Rome, Nick |
| author_facet | Kumaraswamy, V. Vinay Rome, Nick |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_04234 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a conjecture of Wooley and lower bounds for cubic hypersurfaces Kumaraswamy, V. Vinay Rome, Nick Number Theory 11D45, 11P55, 14G05, 11D25, 11N36 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. |
| title | On a conjecture of Wooley and lower bounds for cubic hypersurfaces |
| topic | Number Theory 11D45, 11P55, 14G05, 11D25, 11N36 |
| url | https://arxiv.org/abs/2405.04234 |