Abstract
Very general hypersurfaces in ℝ4 contain ≪r2+(4/9) integer points in any ball of radiusr>1. As a consequence, an irreducible algebraic hypersurface in ℝn (wheren≥4) which is not a cylinder and is of degreed, contains ≤c(d, n)rn−1−(5/9) integer points in a ball of radiusr. This improves on the known boundc(d, n)rn−(3/2).
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