Counting rational points on hypersurfaces and higher order expansions

We study the number of representations of an integer n=F(x) by a homogeneous form in sufficiently many variables. This is a classical problem in number theory to which the circle method has been successfully applied to give an asymptotic for the number of such representations where the integer vector x is restricted to a box of side length P for P sufficiently large. In the special case of Waring's problem, Vaughan and Wooley have recently established for the first time a higher order expansion for the corresponding asymptotic formula. Via a different and much more general approach we derive a multi-term asymptotic for this problem for general forms F(x) and give an interpretation for the occurring lower order terms. As an application we derive higher order expansions for the number of rational points of bounded anticanonical height on the projective hypersurface F(x)=0 for forms F(x) in sufficiently many variables. The main term of this expansion is the one predicted by Manin's conjecture. Our new result gives some evidence for how the conjecture could be refined to cover lower order terms in the setting of high-dimensional complete intersections in projective space