has a solution in Fn. The representation problem of quadratic forms is to determine, in an effective manner, the set of elements of F that are represented by a particular quadratic form over F . We shall discuss the case when F is a field of arithmetic interest, for instance, the field of complex numbers C, the field of real numbers R, a finite field F, and the field of rational numbers Q. The ...

We determine the indexes of all orthogonal Grassmannians of a generic 16dimensional quadratic form in I q . This is applied to show that the 3-Pfister number of the form is ≥ 4. Other consequences are: a new and characteristic-free proof of a recent result by Chernousov–Merkurjev on proper subforms in I q (originally available in characteristic 0) as well as a new and characteristic-free proof ...

The theory of quadratic forms can be regarded as studying an important special case of the general problem of birational classification of algebraic varieties. A typical example of a Fano fibration in minimal model theory is a quadric bundle, which can be viewed without loss of information as a quadric hypersurface over the function field of the base variety. This description feeds the problem ...