Integer points on hypersurfaces

Very general hypersurfaces in ⁴ contain r ^2+(4/9) integer points in any ball of radiusr>1. As a consequence, an irreducible algebraic hypersurface in ⁿ (wheren4) which is not a cylinder and is of degreed, contains c(d, n)r ^{n–1–(5/9)} integer points in a ball of radiusr. This improves on the known boundc(d, n)r ^n–(3/2).Meinem verehrten Lehrer Professor E. Hlawka zum siebzigsten Geburtstag gewidmetWritten with partial support from NSF-MCS-8211461.