Forms in many variables 论文

摘要

Abstract It is proved that if f is a homogeneous form of degree d with rational coefficients then the variety V:f = 0 certainly has rational points if it has non-singular real points and nonsingular p-adic points for every p, and if its singular locus has codimension sufficiently large compared with the degree d. The methods used are derived from those of Davenport (1959); considerable generalizations are made, and geometric conditions have to be introduced. The discussion of the singular integral presents unexpected difficulty.