  Let F be a number field, and let F\subset K be a field extension of degree n.
Suppose that we are given 2r sufficiently general linear polynomials in r
variables over F. Let X be the variety over F such that the F-points of X
bijectively correspond to the representations of the product of these
polynomials by a norm from K to F. Combining the circle method with descent we
prove that the Brauer-Manin obstruction is the only obstruction to the Hasse
principle and weak approximation on any smooth and projective model of X.
