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August 17, 2025Open Access

Counting Rational Points on Smooth Hypersurfaces with High Degree

Puntos clave

Rational points of bounded height on smooth hypersurfaces are shown to have cardinality O(B^(n-2+ε)) in certain conditions.
Specific bounds are improved for smooth projective hypersurfaces in dimensions n ≥ 4 with degree d ≥ 50.
Further improvements are noted for cases where the degree is d ≥ 6, enhancing the dimension growth conjecture.
An analogous result is established for affine hypersurfaces whose projective closure remains smooth.

Resumen

Abstract Let X be a smooth projective hypersurface defined over Q. We provide new bounds for rational points of bounded height on X. In particular, we show that if X is a smooth projective hypersurface in P^n with n 4 and degree d 50, then the set of rational points on X of height bounded by B have cardinality O₍, ₃, (B^n-2+). If X is smooth and has degree d 6, we improve the dimension growth conjecture bound. We achieve an analogue result for affine hypersurfaces whose projective closure is smooth.

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