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arXiv · 2512.03490

Counting rational points on affine hypersurfaces

Anders Mah
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Abstract

We prove an upper bound for the number of rational points of bounded height on irreducible affine hypersurfaces. More precisely, given an irreducible polynomial $f \in \mathbb{Z}[X_1, \dots, X_n]$, we prove an upper bound on the number of points $(x_1, \dots, x_n) \in \mathbb{Q}^n$ such that $f(x_1, \dots, x_n) = 0$ and each component has height at most $B$. To prove this, we require a quantitative form of Hilbert's irreducibility theorem, where we bound the number of reducible specialisations of an irreducible polynomial at rational points of bounded height.

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Anders Mah. 2025-12-03. Counting rational points on affine hypersurfaces. https://arxiv.org/abs/2512.03490

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