Norm forms for arbitrary number fields as products of linear polynomials

Given a number field K/Q and a polynomial P e Q [t], all of whose roots are Q, let X be the variety defined by the equation NK (x) = P (t). Combining additive combinatiorics with descent we show that the Brauer-Manin obstruction is the only obstruction to the Hesse principle and weak approximation on any smooth and projective model of X.