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.
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