CUBIC FORMS OVER IMAGINARY QUADRATIC NUMBER FIELDS AND PAIRS OF RATIONAL CUBIC FORMS

. We show that every cubic form with coefficients in an imaginary quadratic number field K/Q in at least 14 variables represents zero nontrivially. This builds on the corresponding seminal result by Heath-Brown for rational cubic forms. As an application we deduce that a pair of rational cubic forms has a non-trivial rational solution provided that s >= 627. Furthermore, we show that every rational cubic hypersurface in at least 33 variables contains a rational line, and that every rational cubic form in at least 33 variables has "almost-prime" solutions.