Counting Rational Points on Smooth Hypersurfaces with High Degree

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 Pⁿ with n 4 and degree d 50, then the set of rational points on X of height bounded by B have cardinality On,d,ε(Bn-2+ε). 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.